FOURIER AND SPECTRAL MULTIPLIERS IN RN AND IN THE

Transcript

FOURIER AND SPECTRAL MULTIPLIERS
IN RN AND IN THE HEISENBERG GROUP
prof. Fulvio Ricci
A.A. 2003-2004
1
2
Contents
Chapter I Self-adjoint operators and spectral analysis
1.
2.
3.
4.
Review of notions from spectral theory
The Heisenberg sub-Laplacian
The spectral analysis of L1
The joint spectrum of L and i−1 T
Chapter II Mihlin-Hörmander multipliers for constant coefficient operators
1.
2.
3.
4.
5.
6.
7.
Spectral and Fourier Lp -multipliers
The Hardy-Littlewood maximal function
Calderón-Zygmund operators on spaces of homogeneous type
Integral Lipschitz conditions
Non-isotropic dilations in Rn and Calderón-Zygmund kernels
Mihlin-Hörmander conditions on Fourier multipliers
Applications
Chapter III Littlewood-Paley theory and Marcinkiewicz multipliers
1.
2.
3.
4.
Square functions
Littlewood-Paley functions
Marcinkiewicz multipliers
Applications
Chapter IV Fourier analysis on the Heisenberg group
1.
2.
3.
4.
5.
6.
The Heisenberg group
The group Fourier transform
Fourier multipliers
Radial functions and diagonal multipliers
Radiality in Hn
Applications
Chapter V Spectral multipliers of the sub-Laplacian
1. The heat kernel on Hn
2. Smooth multipliers and Schwartz kernels
3. Mihlin-Hörmander multipliers of L
SELF-ADJOINT OPERATORS
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CHAPTER I
AND SPECTRAL ANALYSIS
1. Review of notions from spectral theory
Self-adjoint operators.
We sketch some basic facts about the spectral theory of (possibly unbounded)
self-adjoint operators on a Hilbert space. For a complete treatment, the reader can
consult, e.g. M. Reed, B. Simon Methods of Modern Mathematical Physics, vol. I,
Functional Analysis, vol. II, Fourier Analysis, Self-adjointness. We shall refer to
these books as [RS1, RS2] respectively.
Let H be a Hilbert space, and let T be a linear operator with domain D dense
in H and with values in H.
Let D 0 consist of those elements u ∈ H for which an element w exists such that1
hT v|ui = hv|wi
for all v ∈ D. This w is unique because D is dense, and it is denoted by T ∗ u. The
operator T ∗ defined on D 0 is called the adjoint operator of T .
T is called symmetric if D ⊆ D 0 and T ∗ |D = T . It is called self-adjoint if it is
symmetric and D 0 = D. It is a known fact that a self-adjoint operator is closed
(i.e. its graph in H × H is closed). It follows from the closed graph theorem that
a closed densely defined operator T is bounded if and only if D = H.
The most notable example of a self-adjoint operator is the following. Take H =
L (X, µ), with (X, µ) a measure space. Given an a.e. finite measurable real-valued
function ϕ on X, define
Tϕ f (x) = ϕ(x)f (x) ,
2
on D = {f ∈ L2 (X, µ) : ϕf ∈ L2 (X, µ)}. That D is dense follows from the fact
that, if we denote by Xn the set where ϕ(x) < n, then D contains all L2 -functions
supported on Xn .
Given g ∈ L2 (X, µ), we look for another L2 -function h such that
Z
ϕ(x)f (x)g(x) dµ(x) =
X
Z
f (x)h(x) dµ(x) ,
X
1 We
adopt the following notation: h | i denotes the Hermitean product on a Hilbert space,
whereas h , i denotes a bilinear pairing, e.g. between a distribution and a test function.
Typeset by AMS-TEX
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CHAPTER I
for every f ∈ D. This is possible if and only if h(x) = ϕ(x)g(x) a.e. Hence D 0 = D
and then it is pretty obvious that Tϕ∗ = Tϕ .
Observe that Tϕ is bounded if and only if ϕ ∈ L∞ (X, µ).
The following statement is easy to prove.
Proposition 1.1. Let A : H → H 0 be a unitary transformation between Hilbert
spaces. Given a densely defined operator T on H with domain D, let T 0 be the
operator on H 0 , with domain D 0 = AD, given by T 0 = AT A−1 . Then T is selfadjoint if and only if T 0 is.
In this case we say that T and T 0 are unitarily equivalent.
As a direct consequence of self-adjointness of Tϕ for ϕ real-valued, we have the
following statement for constant coefficient differential operators.
Theorem 1.2. Let P be a polynomial in n variables with real coefficients, and
consider the differential operator L = P (i−1 ∂) = P (i−1 ∂x1 , . . . , i−1 ∂xn ) on Rn .
With H = L2 (Rn ), take
(1.1)
D = {f ∈ L2 (Rn ) : Lf ∈ L2 (Rn )} ,
as the domain of L. Then L is self-adjoint.
Proof. Let A be the unitary transformation of L2 (Rn ) onto itself given by the
Fourier transform multiplied by (2π)−n/2 . Then L is unitarily equivalent to TP .
Since P has real coefficients, TP is self-adjoint. The conclusion follows from Proposition 1.1. The multiplication operator Tϕ described above is more than just an example.
The following statement is proved in [RS1].
Theorem 1.3 (Spectral Theorem, version 1). Let H be a separable Hilbert
space and T be a self-adjoint operator on it with domain D. Then T is unitarily
equivalent to a multiplication operator Tϕ , with ϕ measurable, a.e. finite, and realvalued on some finite measure space (X, µ).
The resolvent set of a closed operator T on a HIlbert space H is defined as the
set of those λ ∈ C for which λI − T has a bounded inverse. The complement of the
resolvent set is the spectrum of T , denoted by σ(T ). The resolvent set is open and
the spectrum is closed.
If Tϕ is as above, it is simple to show that λ is in the resolvent set if and only if
(λ − ϕ)−1 ∈ L∞ (X, µ), i.e. if and only if there is δ > 0 such that
µ{x : |ϕ(x) − λ| < δ} = 0 .
The complement of this set, called the essential range of ϕ, is the spectrum of
Tϕ . Since ϕ is real-valued, clearly σ(Tϕ ) ⊆ R.
It is a general fact that the spectrum of a self-adjoint operator is contained in
R. For separable H, one can appeal to Theorem 1.3, for general H see [RS1].
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Spectral measure.
Definition. A regular projection-valued measure, also called a regular resolution
of the identity, on R is a map E assigning to each Borel subset ω of R an orthogonal
projection E(ω) on some fixed Hilbert space H, satisfying the following properties:
(1) E(∅) = 0 and E(R) = I;
(2) E(ω ∩ ω 0 ) = E(ω)E(ω 0);
(3) if {ωj } is a countable family of pairwise disjoint Borel sets, then
[ X
E
ωj =
E(ωj ) ,
j
j
in the strong topology of L(H);
(4) for every Borel set ω,
E(ω) = sup E(ω 0 ) : ω 0 ⊂ ω , ω 0 compact = inf E(ω 00 ) : ω 00 ⊃ ω , ω 00 open ,
with respect to the partial ordering in the space of bounded self-adjoint operators on H (for which T ≤ T 0 if and only if hT u|ui ≤ hT 0 u|ui for every
u ∈ H).
The support of the measure E is the smallest closed set F such that E(R\F ) = 0.
It follows from (2) that the projections E(ω) commute with each other.
Observe that, if E is a regular resolution of the identity, then
Z
v=
dE(λ)v
R
for every v ∈ H. Also, given v, w ∈ H,
νv,w (ω) = hE(ω)v|wi
is a (scalar-valued) finite Borel measure, and
Z
Z
hv|wi =
dνv,w = hdE(λ)v|wi .
R
R
Clearly, νv,v is positive for every v, and kνv,v k1 = kvk2 . Hölder’s inequality
shows that, for general v, w ∈ H, kνv,w k1 ≤ kukkvk.
Proposition 1.4. Let m be an E-a.e. finite Borel function on R. Then
Z
n
o
D= v∈H:
|m(λ)|2 dνv,v < ∞ ,
R
is dense in H and the operator
Sm v =
Z
m(λ) dE(λ)v
R
is well-defined on D, it commutes with every E(ω), and
Sm E(ω) = Smχω .
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CHAPTER I
Moreover, Sm is bounded on H if and only if m is E-a.e. bounded, and it is
self-adjoint if and only if m is E-a.e. real-valued.
If m1 and m2 are E-a.e. bounded, then Sm1 m2 = Sm1 Sm2 .
Proof. Let Fn be the Borel set where |m(λ)| ≤ n, and let En = E(Fn ). If v ∈ En H,
then νv,v (R \ Fn ) = 0. It follows that v ∈ D.
Since the Fn form an increasing sequence,
from condition (3) that the
S it follows
.
This
is the identity operator,
En converge in the strong topology to E
F
n n
because m is E-a.e. finite. This implies that
[
D̄ ⊆
En H = H .
n
If v ∈ H and m is a simple function,
m(λ) =
X
mj χωj (λ) ,
j
with the ωj pairwise disjoint Borel sets, then
Sm v =
X
mj E(ωj ) ,
j
and
kSm vk2 =
(1.2)
=
X
j,k
X
j
=
Z
R
mj m̄k hE(ωj )v|E(ωk )vi
|mj |2 hE(ωj )v|vi
|m(λ)|2 dνv,v (λ) .
A limiting argument shows that Sm v is well defined for v ∈ D and that (1.2)
holds for every v ∈ D. It also shows that Sm commutes with the E(ω) and that
Sm E(ω) = Smχω .
In particular, this implies that each subspace En H is mapped into itself by Sm .
That the boundedness of Sm is equivalent to the E-a.e. boundedness of m follows
from (1.2). We show now that D is also the domain of the adjoint of Sm .
If u, v ∈ D, then
hSm v|ui =
=
Z
Z
m(λ) dνv,u (λ)
R
m(λ) dνu,v (λ)
R
= hSm u|vi
= hv|Sm ui .
∗
This shows that Sm
extends Sm , i.e. Sm is symmetric.
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∗
Take now u in the domain of Sm
. Then there is w ∈ H such that hSm v|ui = hv|wi
for every v ∈ D. This is equivalent to saying that
hSm v|ui ≤ Cu kvk
(1.3)
for some constant Cu and every v ∈ D.
Let un = En u. Then un ∈ D and un → u. Also, Sm un is also in D. We can
then apply (1.3) with v = Sm un . We find that
2
kSm un k2 = hSm
un |un i
2
= hSm
un |En ui
2
= hSm
un |ui
≤ Cu kSm un k .
Hence kSm un k ≤ Cu for every n. From
2
2
kSm un k = kSmχFn uk =
Z
Fn
m(λ)2 dνu,u (λ) ≤ Cu ,
we obtain that u ∈ D.
Finally, the identity Sm1 m2 = Sm1 Sm2 is trivial if m1 , m2 are simple functions,
and the general case follows by approximation. The next theorem, the proof of which can be found in [RS1], says that resolutions
of the identity are in 1-1 correspondence with self-adjoint operators.
Theorem 1.5 (Spectral Theorem, version 2). Let T be a self-adjoint operator
on a Hilbert space H. Then there is one and only one regular resolution of the
identity of H, {E(ω)}, on R such that
Z
T =
λ dE(λ) .
R
The measure E is called the spectral measure of T . Its support coincides with
σ(T ).
We describe the resolution of the identity of L2 (X, µ) associated with a multiplication operator Tϕ , ϕ being real-valued and a.e. finite.
Given a Borel set ω ⊆ R, let
E(ω)f = f χϕ−1 (ω)
be the orthogonal projection onto the subspace of L2 -functions on X supported on
ϕ−1 (ω). This is clearly a regular resolution of the identity. For f ∈ L2 (X, µ), we
have
Z
νf,f (ω) =
|f (x)|2 dµ(x) .
ϕ−1 (ω)
P
If g(λ) = j gj χωj (λ) is a simple function on R, with the ωj pairwise disjoint,
then
Z
Z
X Z
2
g(λ) dνf,f (λ) =
gj
|f (x)| dµ(x) =
g ϕ(x) |f (x)|2 dµ(x) .
R
j
ϕ−1 (ωj )
X
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CHAPTER I
This identity can be easily extended to more general g.
Consider
Z
S=
λ dE(λ) .
R
Hence the domain D of S consists of the functions f such that
Z
2
λ dνf,f (λ) =
R
Z
X
ϕ(x)2 |f (x)|2 dµ(x) < ∞ ,
i.e. those for which ϕf ∈ L2 (X, µ).
This last example gives the spectral resolution for self-adjoint constant coefficient
differential operators on Rn .
Proposition 1.6. Let L = P (i−1 ∂), where P a polynomial in n variables with real
coefficients, with the domain given in (1.1). Then
L=
Z
λ dE(λ) ,
R
where, denoting by F the Fourier transform,
E(ω)f = F −1 (fˆχP −1 (ω) ) .
The notion of spectral measure associated to a self-adjoint operator T allows to
develop a functional calculus on T , i.e. to define other operators expressed in terms
of the spectral measure, hence depending on T itself.
Let dE(λ) be the spectral measure of T . If m is a Borel measurable function on
R, E-a.e. finite, define
Z
m(T ) =
m(λ) dE(λ) .
R
The function m is called a spectral multiplier.
Extensions of symmetric operators.
The general question if a symmetric operator admits a self-adjoint extension,
and if this extension is unique, requires a detailed study, which is out of the scope
of these notes. The answer is that self-adjoint extensions do not always exist, and
if they exist, they need not be unique. The only fact we want to mention concerns
positive operators.
A symmetric operator T with domain D is called positive if
hT f |f i ≥ 0
for every f ∈ D.
A well-known fact is that every positive symmetric operator admits at least one
self-adjoint extension (see [RS2] for the construcion of a canonical extension, called
the Friedrichs extension).
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For our purposes, it is important to mention the following application. Let
X1 , . . . , Xm be first-order operators on Rn , and denote by Xj0 the formal adjoint of
Xj , i.e.
Xj f (x) =
n
X
ajk (x)∂xk f (x) + a0 (x)f (x) ,
k=1
n
X
Xj0 f (x) = −
k=1
∂xk āj,k (x)f (x) + ā0 (x)f (x) .
We shall assume that
are defined and smooth on all of Rn .
Pmthe coefficients
The operator L = j=1 Xj0 Xj , initially defined on D0 = D(Rn ) is symmetric,
because its adjoint is an extension of L itself, defined on
D = f ∈ L2 (Rn ) : Lf ∈ L2 (Rn ) ,
(with Lf understood in the sense of distributions). It is also clear that L is positive.
Theorem 1.7. The operator L with domain D is self-adjoint, and it is the only
self-adjoint operator, with domain containing D0 , and equal to L on D0 .
Proof. The first part of the statement follows from the Friedrichs construction. For
the second part2 , let (L, D 0 ) be a self-adjoint extension of (L, D0 ). If f ∈ D 0 , by
self-adjointness,
hf |Lgi = hLf |gi
for all g ∈ D 0 . If we restrict to g ∈ D0 ,this implies that h = Lf in the sense of
distributions, so that f ∈ D.
Hence (L, D 0 ) ⊆ (L, D). Passing to the adjoints, the inclusions are reversed; but
both operators are self-adjoint, hence D 0 = D. Spectral analysis of commuting self-adjoint operators.
If T1 , T2 are bounded self-adjoint operators on H, and T1 T2 = T2 T1 , then any
two projections E1 (ω) and E2 (ω 0 ) in the corresponding resolutions of the identity
also commute with each other (see [RS1]).
If we try to extend this statement to unbounded operators, we meet several
difficulties. First of all, the composition T1 T2 is well defined only on those elements
v in the domain of T2 such that T2 v is in the domain of T1 . These elements can
form a very small space, and this space may not coincide with the one constructed
by interchanging the rôle of T1 and T2 . Worse than that, even though the equality
T1 T2 = T2 T1 holds on a dense subspace, the projections in the two resolutions of
the identity need not commute.
We are so led to impose the following definition.
Definition. Let T1 , T2 be self-adjoint operators on H. We say that T1 and T2 commute if the operators {E1 (ω)} and {E2 (ω)} forming the corresponding resolutions
of the identity all commute with each other.
We state without proof the following fact (see [RS1] for the proofs).
2 Instead
of using different symbols, like L0 or others for extensions of the “initial” L, we prefer
to keep the same symbol, specifying the domain whenever there is ambiguity. We shall also write
(L, D1 ) ⊆ (L, D2 ) to denote that the second operator is an extension of the first from the domain
D1 to the domain D2 .
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CHAPTER I
Theorem 1.8. Let T1 , T2 be self-adjoint operators. The following are equivalent:
(i) T1 and T2 commute;
(ii) if λ, µ are non-real numbers, (λI − T1 )−1 and (µI − T2 )−1 commute;
(iii) for every s, t ∈ R, eisT1 and eitT2 commute.
Observe that the operators in (ii) and (iii) are bounded, so that “commutation”
is meant in the ordinary sense.
If T1 and T2 commute, we can define a joint spectral measure on R2 , by defining
E(ω × ω 0 ) = E1 (ω)E2 (ω 0 ) ,
and extending E to every other Borel set in R2 so that conditions (3) and (4) are
satisfied.
Projecting E onto each component, we find E1 and E2 respectively. Hence
T1 =
Z
λ dE(λ, µ) ,
R2
T1 =
Z
µ dE(λ, µ) .
R2
A joint spectral multiplier of T1 and T2 is an E-a.e. finite Borel measurable
function m(λ, µ), and one defines
m(T1 , T2 ) =
Z
m(λ, µ) dE(λ, µ) .
R2
The extension of these notions to a larger number of mutually commuting selfadjoint operators is trivial. Proposition 1.4 remains true for multipliers of more
than one operator.
The support of E is called the joint spectrum of T1 and T2 , denoted by σ(T1 , T2 ).
In contrast with what happens when taking tensor products of scalar-valued measures, it may happen that the support of E is strictly contained in the product of
the support of the Ej ; in other words, the joint spectrum σ(T1 , T2 ) can be strictly
smaller than σ(T1 ) × σ(T2 ).
What can happen is that E1 (ω) and E2 (ω 0 ) are non-zero, but their product is
zero.
Consider the case H = C3 , and

1 0
T1 =  0 1
0 0

0
0  ,
−1

2
T2 =  0
0

0 0
3 0 .
0 3
Then σ(T1 ) = {1, −1} and σ(T2 ) = {2, 3}. The spectral projections have the
following ranges:
E1 ({1})
span (e1 , e2 )
E1 ({−1})
span (e3 )
E2 ({2})
span (e1 )
E2 ({3})
span (e2 , e3 ) .
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Hence the only non trivial products are E(1, 2), E(1, 3), E(−1, 3), so that
σ(T1 , T2 ) = {(1, 2), (1, 3), (−1, 3)} .
A more interesting example is the following. On Rn , take T1 = i−1 ∂x1 and
T2 = ∆, the Laplacian. The domains are those described before, making each of
them self-adjoint.
By Proposition 1.6, if ω, ω 0 are Borel subsets of R,
E1 (ω)f = F −1 (fˆχ{ξ:ξ1 ∈ω} ) ,
E2 (ω 0 )f = F −1 (fˆχ{ξ:|ξ|2 ∈ω 0 } ) .
This implies that E(ω × ω 0 ) = 0 if ω × ω 0 does not intersect the epi-parabola
µ ≥ λ2 , and one can show easily that the joint spectrum is the full epi-parabola.
Fourier multipliers.
On Rn take Tj = i−1 ∂xj , for 1 ≤ j ≤ n. One can easily verify that the joint
spectrum is all of Rn , and that, if m(ξ) is an a.e. finite Borel function on Rn , then3
m(i−1 ∂)f (x) = F −1 (fˆm)(x) .
Hence joint spectral multipliers for the system i−1 ∂ coincide with the Fourier
multipliers.
Similarly, if m is an a.e. finite Borel function on R, then
m(∆)f (x) = F −1 fˆm(| · |2 ) (x) ,
i.e. the spectral multipliers of ∆ coincide with the radial Fourier multipliers.
2. The Heisenberg sub-Laplacian
In this section we present what will be for us the main example of an operator
of the form described in Theorem 1.7. The group-theoretic notions connected with
the operators below are postponed to a future section.
We take R2n+1 = Rn × Rn × R with coordinates (x, y, t), and define 2n vector
fields X1 , . . . , Xn , Y1 , . . . , Yn as
(2.1)
X j = ∂ xj −
yj
∂t ,
2
Yj = ∂ y j +
xj
∂t .
2
Then Xj0 = tXj = −Xj , Yj0 = tYj = −Yj , so that the Heisenberg sub-Laplacian
(2.2)
L=
n
X
j=1
(Xj0 Xj
+
Yj0 Yj )
n
X
=−
(Xj2 + Yj2 )
j=1
is positive on L2 (R2n+1 ), and self-adjoint on the domain
D = f ∈ L2 (R2n+1 ) : Lf ∈ L2 (R2n+1 ) .
3 We
use the symbol ∂ to denote the system (∂x1 , . . . , ∂xn ) in short.
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CHAPTER I
Notice that S(R2n+1 ) ⊂ D. The explicit expression of L,
n
|x|2 + |y|2 2 X
(xj ∂yj − yj ∂xj )∂t ,
L = ∆ x + ∆y −
∂t +
4
j=1
shows that L is not elliptic (e.g., L = ∆x + ∆y at the origin). It is however
hypoelliptic, according to Hörmander’s theorem4 . This follows from the fact that,
for every j,
[Xj , Yj ] = ∂t ,
and the system of vector fields {X1 , Y1 , . . . , Xn , Yn , ∂t } gives a basis of the tangent
space at each point of R2n+1 .
The constant vector field ∂t is usally denoted by T .
In order to work out its spectral decomposition, it is preferable to replace L by
another operator, unitarily equivalent to it.
Denote by Ft the partial Fourier transform in R2n+1 in the variable t. Then
A = (2π)−1/2 Ft is a unitary operator on L2 (R2n+1 . We set
L̃ = ALA−1 = Ft LFt−1 .
Then L̃ is self-adjoint, with domain
D̃ = g ∈ L2 (R2n+1 ) : L̃g ∈ L2 (R2n+1 ) .
If we perform the same conjugation by Ft on the Xj and Yj , we obtain
(2.3)
λ
X̃j = Ft Xj Ft−1 = ∂xj − i yj ,
2
λ
Ỹj = Ft Yj Ft−1 = ∂yj + i xj ,
2
and
(2.4)
L̃ =
n
X
j=1
(X̃j0 X̃j + Ỹj0 Ỹj ) = −
n
X
(X̃j2 + Ỹj2 ) .
j=1
It must be observed that derivatives are only taken in the variables xj , yj , and not
in λ. We shall also regard the first-order operators in (2.3) as acting on functions
of (x, y) ∈ R2n , taking λ as a parameter.
When we do so, we call Xλ,j , Yλ,j the operators in (2.3), and Lλ the operator in
(2.4). If we set g λ (x, y) = g(x, y, λ), this means that
L̃g(x, y, λ) = Lλ g λ (x, y) .
By Theorem 1.7, Lλ is self-adjoint on L2 (R2n ) for every λ, with domain
Dλ = f ∈ L2 (R2n ) : Lλ f ∈ L2 (R2n ) .
For λ = 0, Lλ = ∆; for λ 6= 0, Lλ is called the twisted Laplacian. The following
statement is obvious.
4 See
the notes of the course “Sub-Laplacians on nilpotent Lie groups”.
13
Lemma 2.1. A function g(x, y, λ) ∈ L2 (R2n+1 ) belong to D̃ if and only if g λ ∈ Dλ
for a.e. λ and
Z
kLλ g λ k22 dλ < ∞ .
R
We describe the spectral measure of L̃.
Proposition 2.2. Denote by Ẽ(ω) the L̃-spectral measure of a Borel subset ω of
R, and by Eλ (ω) its Lλ -spectral measure. Then
Ẽ(ω)g(x, y, λ) = Eλ (ω)g λ (x, y) .
Proof. One easily checks that Ẽ is a regular projection-valued measure. Then we
just need to identify the operator
Z
A=
µ dẼ(µ)
R
together with its domain.
Setting
νg λ ,hλ (ω) = hEλ (ω)g λ |hλ i ,
νg,h (ω) = hẼ(ω)g|hi ,
we have
νg,g (ω) =
Z
λ
R
λ
hEλ (ω)g |g i dλ =
Z Z
R
dνg λ ,g λ (µ) dλ .
ω
Hence the domain of A consists of the functions g such that
Z
Z
2
µ2 dνg λ ,g λ (µ) dλ < ∞ .
µ dνg,g (µ) =
R2
R
But this means that g λ ∈ Dλ for a.e. λ and that
Z
kLλ g λ k22 dλ < ∞ ,
R
i.e. g is in the domain of L̃. Moreover,
Z
hAg|hi =
µ dνg,h (µ)
R
Z
=
µ dνg λ ,hλ (µ) dλ
R2
Z
= hLλ g λ , hλ i dλ
R
= hL̃g|hi .
In view of Proposition 2.2, it will be interesting to obtain an explicit description
of the spectral measures Eλ . The case λ = 0 is very simple, because L0 is the
Laplacian, but of course we are more interested in λ 6= 0.
The following lemma shows that the operators Lλ with λ 6= 0 can be conjugated
among themselves, up to a constant factor, by unitary operators.
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CHAPTER I
Lemma 2.3. For s > 0, let As be the unitary operator As f (x, y) = sn f (sx, sy) on
L2 (R2n ). Then
L±s2 = s2 As L±1 A−1
s .
If Bf (x, y) = f (x, −y), then
L−λ = BLλ B −1 .
If Fλ is the spectrum of Lλ , and λ 6= 0,
Fλ = |λ|F1 = |λ|µ : µ ∈ F1 .
Finally, if m(µ) is a spectral multiplier and λ 6= 0, and ε = 0, 1 depending on
whether λ is negative or positive,
m(Lλ ) = B ε A 21 m |λ|L1 A−1 1 B −ε .
|λ|
|λ| 2
Proof. Given a Schwartz function f on R2n , it follows from (2.3) that
λ
Xλ,j (As f )(x, y) = sn+1 ∂xj f (sx, sy) − i sn yj f (sx, sy)
2
n+1
=s
Xλ/s2 ,j f (sx, sy)
= sAs (Xλ/s2 ,j f )(x, y) .
It follows that As Dλ/s2 = Dλ and
Lλ As = s2 As Lλ/s2 .
This gives the first assertion, and the second can be proved in a similar and
simpler way.
By uniqueness of the spectral measure associated with a self-adjoint operator,
setting C = B ε A 12 , from the identity
|λ|
Z
µ dEλ (µ) = Lλ
R
= |λ|CL1 C −1
Z
= |λ|C
µ dE1 (µ) C −1
Z R
−1
=C
µ dE1 |λ| µ C −1
R
we derive that
Eλ (ω) = CE1 |λ|−1 ω C −1 .
Hence the support Fλ of Eλ and the support F1 of E1 are in the stated relation.
To conclude,
Z
m(Lλ ) =
m(µ) dEλ (µ)
R
Z
−1
=C
m(µ) dE1 |λ| µ C −1
ZR
=C
m(|λ|µ) dE1 µ C −1
R
= Cm |λ|L1 C −1 .
15
3. The spectral analysis of L1
In order to complete the analysis, we then have to determine the spectral measure
E1 of L1 . The first remark is that we can reduce our analysis to n = 1, i.e. to the
operator
2 2
i
i
(3.1)
− ∂x − y − ∂y + x
2
2
on R2 .
This reduction is based on the following fact (for notational convenience, we
state it for two operators, the extension to n operator being left to the reader).
Lemma 3.1. Let L1 , L2 be self-adjoint differential operators on L2 (Rd ) with domains Dj = {f : Lj f ∈ L2 (Rd )} and spectra F1 , F2 ⊆ R. On (Rd )2 = R2d , with
coordinates (x1 , x2 ) with xj ∈ Rd , consider the differential operator L acting as
Lf = (L1 )x1 f + (L2 )x2 f ,
in the sense that each Lj acts on the corresponding variable xj ∈ Rd . Then L is
self-adjoint with domain D = {f : Lf ∈ L2 (R2d )}, and with spectrum F = F1 + F2 .
Proof. It is easy to verify that the operators (Lj )xj commute as self-adjoint operators on L2 (R2d ), with domains D̃1 = {f (x1 , x2 ) : f (·, x2 ) ∈ D1 for a.e. x2 } and
similarly for D̃2 . It is also easy to verify that their joint spectrum is the cartesian
product F1 × F2 in R2 . The conclusion follows by applying the spectral multiplier
m(λ, µ) = λ + µ. We prefer to use ad-hoc notations at this stage, and set
i
X = ∂x − y ,
2
i
Y = ∂y + x ,
2
calling L the operator (3.1).
We shall see that L has a discrete spectrum, and we can explicitely construct a
complete system of eigenfunctions. For this construction it is crucial to introduce
the complex operators
1
z̄
(X − iY)f = ∂z f + f
2
4
1
z
Z̄f = (X + iY)f = ∂z̄ f − f .
2
4
Zf =
(3.2)
For reasons that will be immediately clear, we call Z the annihilation operator
and Z̄ the creation operator.
Since
Zf = e−
(3.3)
|z|2
4
∂z (e
|z|2
4
f) ,
Z̄f = e
|z|2
4
∂z̄ (e−
|z|2
4
f) ,
it follows that
Zf = 0 ⇐⇒ f = e−
Z̄f = 0 ⇐⇒ f = e
|z|2
4
|z|2
4
× an antiholomorphic function ,
× a holomorphic function .
16
CHAPTER I
This implies that Z̄ is injective on L2 (C), whereas Z has a big null-space in
L (C). For j, k ∈ N, define
2
hj,0 (z) = z̄ j e−
(3.4)
|z|2
4
,
hj,k (z) = Z̄ k hj,0 (z) .
Proposition 3.2. The functions hj,k , with j, k ∈ N, form a complete orthogonal
system in L2 (C), and
L1 hj,k = (2k + 1)hj,k .
Proof. Since
Z̄Zf =
1
1
(X + iY)(X − iY) = X 2 + Y 2 − i[X , Y] ,
4
4
and [X , Y] = iI by (2.3), we have that
L = −4Z̄Z + I .
(3.5)
But Zhj,0 = 0, hence Lhj,0 = hj,0 . A similar computation shows that
L = −4Z Z̄ − I ,
(3.6)
so that
Lhj,k = (−4Z̄Z + I)Z̄hj,k−1
= Z̄(−4Z Z̄ + I)hj,k−1
= Z̄(L + 2I)hj,k−1 .
By induction, Lhj,k = (2k + 1)hj,k .
It also follows by induction that hj,k equals a polynomial in z, z̄ times e−
particular, hj,k ∈ S(C), hence in the domain of L1 in L2 (C). We then have
(2k + 1)hhj,k |hj 0 ,k0 i = hL1 hj,k |hj 0 ,k0 i
= hhj,k |L1 hj 0 ,k0 i
= (2k 0 + 1)hhj,k |hj 0 ,k0 i ,
so that the two functions are orthogonal if k 6= k 0 .
If k = k 0 = 0, then
(3.7)
hhj,0 |hj 0 ,0 i =
=
Z
C
Z +∞
0
0
z̄ j z j e−
|z|2
2
dz
2
r
j+j 0 +1 − r2
e
dr
Z
2π
ei(j
0
0
−j)θ
dθ ,
|z|2
4
. In
17
which is zero if j 6= j 0 . The same conclusion follows by induction for k = k 0 > 0,
since
hhj,k |hj 0 ,k i = hZ̄hj,k−1 |Z̄hj 0 ,k−1 i
= −hhj,k−1 |Z Z̄hj 0 ,k−1 i
1
= hhj,k−1 |(L + I)hj 0 ,k−1 i
4
k
= hhj,k−1 |hj 0 ,k−1 i .
2
(3.8)
From (3.3) we obtain that
(3.9)
hj,k (z) = e
|z|2
4
∂z̄k z̄ j e−
and, by Leibniz’s formula,
hj,k (z) = e
(3.10)
=
−
(
|z|2
4
min{j,k}
X
(−1)
`=0
k−`
|z|2
2
,
k j(j − 1) · · · (j − ` + 1) j−` k−`
z̄ z
2k−`
`
|z|2
z̄ j−k Pj,k |z|2 e− 4
|z|2
z k−j Pj,k |z|2 e− 4
if j ≥ k ,
if j < k ,
where Pj,k is a polynomial of degree equal to min{j, k}.
This implies that the linear span of the hj,k contains all functions of the form
Q(z, z̄)e−
|z|2
4
, where Q is an arbitrary polynomial in two variables. Switching back
to real coordinates, we find that for every m, n the function xm y n e−
linear span of the hj,k .
Assume that f ∈ L2 (R2 ) is orthogonal to all the hj,k . Then
(3.11)
Z
f (x, y)xm y n e−
x2 +y 2
4
x2 +y 2
4
is in the
dx dy = 0 ,
R2
for all m, n. Consider the functions g(x, y) = f (x, y)e−
G(ζ1 , ζ2 ) =
Z
x2 +y 2
4
and
g(x, y)e−i(xζ1 +yζ2 ) dx dy .
R2
Because of the rapid decay of g at infinity due to the Gaussian factor, G is
defined on all of C2 , and holomorphic. Hence ĝ = G|R2 , i.e. the Fourier transform
of g, is real-analytic on all R2 . Condition (3.11) says that all derivatives of ĝ at the
origin are zero. Hence ĝ = 0, i.e. g = 0 and finally f = 0. This shows that L has a discrete spectrum, the eigenvalues being the odd positive
integers. Combining this with Lemma 3.1 and with Lemma 2.3, we obtain the
following description of the spectral measure Eλ of Lλ on R2n .
18
CHAPTER I
Corollary
3.3. Let n = 1. If λ 6= 0, the spectrum of Lλ , as an operator on L2 (R2 ),
is |λ|(2k + 1) : k ∈ N .
√ If λ > 0, Eλ {λ(2k + 1)} is the orthogonal projection onto span hj,k λ z :
j∈N .
p
|λ| z̄ :
If λ < 0, Eλ {|λ|(2k +1)} is the orthogonal projection onto span hj,k
j∈N .
For general n, the spectrum of Lλ is |λ|(2k + n) : k ∈ N . If λ > 0, a complete
orthogonal system of eigenfunctions for the eigenvalue λ(2k + n) is given by the
products
n
Y
√
hji ,ki λ zi ,
i=1
with ji , ki ∈ N and k1 + · · · + kn = k (and similarly for λ < 0, replacing λ by |λ|
and zi by z̄i ).
One may observe that the coefficient in (3.10)
(−1)
k−`
(−1)k−` k!j!
k
j(j − 1) · · · (j − ` + 1) =
`
`!(k − `)!(j − `)!
exhibits a symmetry in j and k, which give the identity
(3.12)
hj,k (z) = (−2)j−k hk,j (z̄) = (−2)j−k hk,j (z) .
Modulo normalizations, the polynomials Pj,k appearing in (3.10) are the so-called
(α)
Laguerre polynomials Lm , with α = |j − k| and m = min{j, k}. The Laguerre
polynomial belong to the class of confluent hypergeometric functions, and they are
described, e.g., in the book Higher transcendental functions, vol. II, by A. Erdelyi,
W. Magnus, F. Oberhettinger, F. Tricomi.
4. The joint spectrum of L and i−1 T
We have already observed in Section 2 that
T = ∂t = [Xj , Yj ] ,
this last equality holding for every j.
Since T has constant coefficients, T Lf = LT f for every f ∈ D(R2n+1 ). It takes
a little thought to realize that, as self-adjoint operators, L and i−1 T commute.
Proposition 4.1. L and i−1 T commute with each other. The joint spectrum of L
and i−1 T consist of the pairs λ, |λ|(2k + n) with λ ∈ R and k ∈ N, and of the
pairs (0, µ) with µ ≥ 0.
Proof. It is convenient to replace the two operators by the unitarily equivalent
1
ones obtained by conjugating both of them with (2π)− 2 Ft . Instead of L, we then
consider L̃, introduced in Section 2, and instead of i−1 T the multiplication operator
Sf (x, y, λ) = λf (x, y, λ) .
19
We call Ẽ and Ẽ 0 the spectral measures of L̃ and S respectively. The example
given in Section 1 shows that
Ẽ 0 (ω 0 )f (x, y, λ) = χω 0 (λ)f (x, y, λ) ,
and, by Proposition 2.2, calling f λ (x, y) = f (x, y, λ),
Then
(4.1)
Ẽ(ω)f (x, y, λ) = Eλ (ω)f λ (x, y) .
Ẽ(ω)Ẽ 0 (ω 0 )f (x, y, λ) = Ẽ 0 (ω 0 )Ẽ(ω)f (x, y, λ) = χω 0 (λ) Eλ (ω)f λ (x, y) ,
hence L and i−1 T commute.
In order to determine the joint spectrum of L̃ and S, we discuss the possibility
that
(4.2)
Ẽ(ω)Ẽ 0 (ω 0 ) = 0 .
We can take ω, ω 0 open intervals; for the moment we assume that 0 6∈ ω 0 , say
ω 0 ⊂ R+ , to fix the notation.
By (4.1), (4.2) happens if Eλ (ω) = 0 for every λ ∈ ω 0 . We show that the converse
is also true.
Let λ0 ∈ ω 0 be such that Eλ0 (ω) 6= 0. Then ω contains a point λ0 (2k + n) for
some k ∈ N. Then there is δ > 0 such that λ(2k + n) ∈ ω for |λ − λ0 | < δ. Let
ω 00 = (λ0 − δ, λ0 + δ) ⊂ ω 0 .
Take h(x, y) a non-trivial
in the (2k +n)-eigenspace of L1 . By Corollary
√
√ function
λ
3.3, h (x, y) = h λ x, λ y is in the λ(2k + n)-eigenspace of Lλ for λ > 0, hence
Eλ (ω)hλ = hλ for λ ∈ ω 00 .
Let f (x, y, λ) = χω 00 (λ)hλ (x, y). By (4.1),
Ẽ(ω)Ẽ 0 (ω 0 )f = f ,
contradicting (4.2).
This proves that a point (λ, µ) with λ 6= 0 is in the joint spectrum if and only
if µ = |λ|(2k + n) for some k ∈ N. Then the joint spectrum must also contain the
points in the closure of this set, i.e. the points (0, µ) with µ ≥ 0.
On the other hand Ẽ is zero on the negative half-line, and this concludes the
proof. The joint spectrum of L and i−1 T is called the Heisenberg fan.
20
CHAPTER I
MIHLIN-HÖRMANDER MULTIPLIERS
21
CHAPTER II
FOR CONSTANT COEFFICIENT OPERATORS
1. Spectral and Fourier Lp -multipliers
Definition. Let {T1 , . . . , Tn } be a commuting system self-adjoint operators, acting
on L2 (X, µ). If 1 ≤ p ≤ ∞, we say that a measurable, a.e. finite, function m on
Rn is a spectral Lp -multiplier if the operator m(T1 , . . . , Tn ) extends to a bounded
operator from Lp (X, µ) to itself.
In particular, By Proposition 1.4 in Chapter I, m is a spectral L2 -multiplier if
and only if it is bounded a.e. with respect to the joint spectral measure of the Tj .
In this chapter we shall take X = Rn , with µ the Lebesgue measure, and we
shall discuss spectral multipliers for
(1) the system i−1 ∂xj , 1 ≤ j ≤ n, i.e. the Fourier multipliers on Rn ;
(2) the Laplacian, i.e. the radial Fourier multipliers;
(3) other constant coefficient operators, satisfying homogeneity conditions that
will be defined below.
We shall mainly restrict ourselves to p 6= 1, ∞. In this section we make some
preliminary considerations, starting with case (1).
Lemma 1.1. If m is a Fourier Lp -multiplier on Rn for some p ∈ (1, ∞), then
m ∈ L∞ (Rn ). The set of points 1/p ∈ (0, 1) such that m is a Fourier Lp -multiplier
is an interval, symmetric w.r. to 1/2.
Proof. Assume that m is real and that Sm = m(i−1 ∂) is bounded on Lp (Rn ). By
Proposition 1.4 in Chapter I, Sm is self-adjoint. For f, g ∈ D(Rn ),
hSm f |gi = hf |Sm gi ,
0
which implies that Sm is also bounded on the dual space Lp (Rn ). By the RieszThorin interpolation theorem, Sm is bounded on Lq (Rn ) for every q between p
and p0 . This proves the second part of the statement. In particular Sm is L2 bounded, hence m ∈ L∞ (Rn ).
If m = m1 + im2 is not real and Sm is Lp -bounded, take f, g ∈ D(Rn ) real. Then
hSm f |gi ≤ hSm f |gi + hSm f |gi ≤ Ckf kp kgkp0 ,
1
2
and this easily implies that both Sm1 and Sm2 are Lp -bounded.
One next remarks concern homogeneity.
Typeset by AMS-TEX
22
CHAPTER II
For r = (r1 , . . . , rn ) ∈ (R+ )n , define
r · x = (r1 x1 , . . . , rn xn )
on Rn , and the dilation operator
δr f (x) = fr (x) = f (r1 x1 , . . . , rn xn ) .
It is easy to verify that, if f is in the domain of i−1 ∂xj , the same is true for fr
and that
(i−1 ∂xj ) ◦ δr = rj δr ◦ (i−1 ∂xj ) .
Hence
i
−1
∂ xj =
Z
ξj dE(ξ)
Z
−1
−1
= δ r ◦ rj
ξj dE(ξ) ◦ δr
Rn
Z
−1
= δr ◦
ξj dE(r · ξ) ◦ δr ,
Rn
Rn
showing that
(1.1)
E(ω) = δr−1 ◦ E(r · ω) ◦ δr ,
for every Borel set ω.
Hence, for every multiplier m,
Z
S mr =
m(r · ξ) dE(ξ)
Rn
Z
=
m(ξ) dE(r−1 · ξ)
Rn
(1.2)
Z
−1
= δr ◦
m(ξ) dE(ξ) ◦ δr
Rn
=
δr−1
◦ Sm ◦ δ r .
For every p, kfr kp = (r1 r2 · · · rn )−1/p kf kp , so that
kSmr f kp = kSm f kp .
This proves the following statement.
Proposition 1.2. If m is a Fourier Lp -multiplier, and r ∈ (R+ )n , then mr is also
a Fourier Lp -multiplier, and the operators Sm and Smr have the same norm.
Spectral multipliers of the Laplacians correspond to a special subclass of the
Fourier multipliers. In fact, if m(λ) is Borel measurable on the positive half-line,
then
(1.3)
m(∆) = m̃(i−1 ∂) ,
with m̃(ξ) = m |ξ|2 . Observe that if E denotes now the spectral measure of ∆ on
the positive half-line, then (1.1) and (1.2) hold for r = (r, . . . , r), with r · ω replaced
by rω. As a consequence, we have the following analogue of Proposition 1.2.
23
Proposition 1.3. If m is a spectral Lp -multiplier of ∆, and r > 0, then mr (λ) =
m(r −1 λ) is also a spectral Lp -multiplier, and the operators Sm and Smr have the
same norm.
This kind of homogeneity argument is quite general. Staying in the context of
R , let L = P (i−1 ∂) be any constant coefficient, self-adjoint differential operator,
and assume that there are positive exponents λ1 , . . . , λn and k such that
n
P (r λ1 ξ1 , . . . , r λn xn ) = r k P (x1 , . . . , xn ) .
The expressions non-isotropic dilations and non-isotropic homogeneity refer to
this more general situation5 .
One example is
X
2
n
2
2
L = −∂x1 +
∂ xj
,
j=2
ξ12
P
n
2
j=2 ξj
2
corresponding to P (ξ) =
+
, with λ1 = 2, λ2 = · · · = λn = 1, and
k = 4.
Again, spectral multipliers m of L correspond
to special Fourier multipliers,
because (1.3) holds with m̃(ξ) = m P (ξ) . In this more general case, (1.1) and
(1.2) hold for r = (r d1 , . . . , r dn ), with r > 0. Proposition 1.3 holds unchanged.
The rest of the chapter is devoted to the presentation of conditions on the multipliers which are invariant under dilations and which assure that they define bounded
operators on Lp for all p ∈ (1, ∞). This requires the introduction of the CalderónZygmund theory of singular integrals.
2. The Hardy-Littlewood maximal function
Il punto di partenza della teoria di Calderón-Zygmund è l’analisi dell’operatore
massimale di Hardy-Littlewood. In questo paragrafo ne presentiamo gli aspetti
che ci consentiranno più avanti di inglobare in un unico contesto lo studio dei
moltiplicatori spettrali di tutti gli operatori differenziali presentati nel Capitolo I.
Sia X un insieme. Una quasi-distanza su X è una funzione d da X × X in R tale
che
(1)
(2)
(3)
(4)
d(x, y) ≥ 0 per ogni x, y ∈ X;
d(x, y) = 0 se e solo se x = y;
d(x, y) = d(y, x) per ogni x, y ∈ X;
esiste una costante c ≥ 1 tale che
d(x, z) ≤ c d(x, y) + d(y, z)
(2.1)
per ogni x, y, z ∈ X.
5 Despite
the name, the isotropic case, λ1 = · · · = λn , is also included.
24
CHAPTER II
Una quasi-distanza induce in modo naturale una topologia su X, una cui base
è costituita dalle palle B(x, r) = {y : d(x, y) < r}. Sia ora m una misura di Borel
positiva su X. Si dice che m è doubling se esiste una costante c0 tale che
(2.2)
m B(x, 2r) ≤ c0 m B(x, r)
per ogni x ∈ X e r > 0.
Definizione. Una terna (X, d, m), dove d è una una quasi-distanza su X e m è
una misura doubling, si dice uno spazio di natura omogenea.
Esempi.
(1) Ovviamente Rn , con la distanza Euclidea e la misura di Lebesgue, è di
natura omogenea.
(2) Anche Z, con la distanza d(n, m) = |n − m| e la misura del conteggio
m(E) = cardE, è di natura omogenea.
(3) Sia α > −n. Allora R, con la distanza Euclidea e la misura dm(x) = |x|α dx,
è di natura omogenea.
(4) Si prenda X = Rn , e siano d1 , . . . , dn > 0. Si ponga
o
n
1/d1
1/dn
.
d(x, y) = max |x1 − y1 |
, . . . , |xn − yn |
Allora d è una quasi-distanza e, con la misura di Lebesgue, Rn è uno
spazio di natura omogenea.
(5) La sfera unitaria S n−1 , dotata della distanza indotta da Rn e della misura
di Hausdorff σ, è uno spazio di natura omogenea.
Sia X uno spazio di natura omogenea.
Definizione. Sia f localmente integrabile rispetto alla misura m. La funzione
Z
1
|f (y)| dy ,
(2.3)
M f (x) = sup
x∈B m(B) B
si chiama funzione massimale di Hardy-Littlewood di f e l’operatore M : f 7−→ M f
operatore massimale di Hardy-Littlewood.
Chiaramente M non è lineare (si osservi che M f (x) ≥ 0 per ogni f ), ma solo
sub-lineare, nel senso che
(2.4)
M (f + g) ≤ M f + M g ,
M (λf ) = |λ|M f .
Lemma 2.1. La funzione M f è semicontinua inferiormente, e dunque misurabile.
Proof. Sia M (x0 ) > α. Esiste allora una palla B contenente x tale che
Z
1
|f (y)| dy > α .
m(B) B
Ma allora per ogni x ∈ B M f (x) > α. MIHLIN-HÖRMANDER MULTIPLIERS
25
Osservazione. La definizione classica di funzione massimale di Hardy-Littlewood
(nel contesto X = Rn , con distanza euclidea e misura di Lebesgue) è la seguente:
Z
1
0
(2.5)
M f (x) = sup
|f (y)| dy ,
r>0 |B(x, r)| B(x,r)
limitandosi quindi a considerare le medie di |f | sulle palle centrare in x. Chiaramente M 0 f (x) ≤ M f (x); tuttavia M 0 f non è necessariamente semicontinua inferiormente. La misurabilità di M 0 f segue dal fatto che la funzione
Z
1
F (x, r) =
|f (y)| dy
|B(x, r)| B(x,r)
è continua in r, per cui l’estremo superiore in (2.5) può essere ristretto a r ∈ Q.
In un generico spazio di natura omogenea, nulla assicura che F (x, r) sia continua
in r, per cui è preferibile ricorrere alla (2.3).
Ci interessa ora discutere la limitatezza di M sugli spazi Lp (X), ossia la validità
di disuguaglianze del tipo
kM f kp ≤ Ckf kp .
Si noti che dalle (2.4) segue che |M f − M g| ≤ M (f − g), e dunque la limitatezza
su Lp equivale alla continuità, come per gli operatori lineari.
Ovviamente M è limitato su L∞ (X). All’altro estremo, per p = 1, non si ha
limitatezza in generale. Per esempio, nella situazione classica (X = Rn ecc.), se f
è la funzione caratteristica della palla unitaria, si vede facilmente che
M f (x) ≥
C
,
1 + |x|n
per cui M f 6∈ L1 .
Tuttavia, M risulta limitato in un senso più debole, fatto che costituisce il punto
fondamentale della teoria di Calderón-Zygmund.
Definizione. Sia T un operatore sub-lineare definito da Lp (X) a Lq (X) (con 1 ≤
p, q < ∞) a valori nelle funzioni misurabili su X. Si dice che T è di tipo debole
(p, q), se per ogni α > 0
q
kf kp
.
(15.3)
|{x : M f (x) > α}| ≤ C
α
Si noti che, se T è limitato da Lp (X) a Lq (X), allora T è di tipo debole (p, q)
per la disuguaglianza di Chebishev.
Dimostreremo per prima cosa che M è di tipo debole (1, 1), basandoci sul
seguente lemma di ricoprimento di Vitali.
Lemma 2.2. In uno spazio di natura omogenea X, sia {Bj }j∈J un ricoprimento
finito di un insieme misurabile E mediante palle. Esiste allora una sottofamiglia
{Bj }j∈J 0 tale che Bj ∩ Bk = ∅ per j, k ∈ J 0 , j 6= k, e inoltre
[
Bj ≥ κ|E| ,
j∈J 0
26
CHAPTER II
dove κ dipende solo dalle costanti c, c0 in (2.1), (2.2).
Proof. Sia Bj1 una palla che abbia misura massima. Induttivamente si prenda
Bjk+1 in modo che abbia misura massima tra le palle disgiunte da Bj1 ∪ · · · ∪ Bjk .
Ovviamente il procedimento si arresta dopo un numero finito di passi, precisamente
quando non ci sono più palle disgiunte da Bj1 ∪· · ·∪Bjk . Poniamo J 0 = {j1 , . . . , jk }.
Data una palla B di raggio r, sia B ∗ la palla con lo stesso centro e raggio 2cr,
dove c è la costante che appare nella (2.1). Si noti che, se due palle B, B 0 hanno
intersezione non vuota e il raggio di B 0 è minore o uguale al raggio di B, allora
B0 ⊆ B∗.
Sia B 0 una delle palle avanzate. Necessariamente essa interseca almeno una di
quelle selezionate. Sia `¯ il primo intero ` tale che B 0 ∩ Bj` 6= ∅. Allora il raggio di
Bj`¯ è maggiore o uguale al raggio di B 0 , e dunque
B 0 ⊆ Bj∗`¯ .
Di conseguenza
E⊆
Se 2ν ≥ 2c, si ha allora
per cui, con κ = c0
[
j∈J
Bj ⊆
[
Bj∗ ,
j∈J 0
ν
m(Bj∗ ) ≤ m B(x, 2k r) ≤ c0 m(B) .
−ν
,
[
−1
−1 ∗
Bj .
|Bj | = κ |E| ≤
|Bj | = κ
j∈J 0 j∈J 0
j∈J 0
X
X
Teorema 2.3. L’operatore M è di tipo debole (1, 1).
Proof. Data f ∈ L1 (X) e dato α > 0, sia Eα = {x : M f (x) > α}. Per il Lemma 2.1,
Eα è aperto e la sua misura è l’estremo superiore delle misure dei suoi sottoinsiemi
compatti.
Sia E un sottoinsieme compatto di Eα . Preso x ∈ E, essendo M f (x) > α, esiste
una palla Bx contenente x tale che
Z
1
|f (y)| dy > α ,
|Bx | Bx
ossia
1
|Bx | ≤
α
Z
Bx
|f (y)| dy .
Per la compattezza di E, si può estrarre da {Bx } un sottoricoprimento finito,
e quindi,Papplicando il Lemma 2.2, un sottoinsieme finito {Bxj } di palle disgiunte
tali che j |Bxj | ≥ κ|E|. Si ha allora
Z
X
κ−1 X
kf k1
−1
.
|E| ≤ κ
|Bxj | ≤
|f (y)| dy ≤ κ−1
α
α
B
x
j
j
j
Di conseguenza |Eα | ≤ κ−1 kf k1 /α.
Avendo a disposizione la limitatezza su L∞ e il tipo debole (1,1), possiamo applicare il seguente Teorema di interpolazione di Marcinkiewicz, la cui dimostrazione
si trova su E.M. Stein, G. Weiss, An introduction to Fourier analysis on Euclidean
spaces.
27
Teorema 2.4. Sia T un operatore sub-lineare che sia di tipo debole (p 0 , q0 ) e di
tipo debole (p1 , q1 ), con 1 ≤ p0 ≤ q0 , 1 ≤ p1 ≤ q1 , p0 6= p1 , q0 6= q1
Allora, dato t ∈ [0, 1] e posto
1−t
t
1
=
+
pt
p0
p1
1−t
t
1
=
+
,
qt
q0
q1
T è di tipo forte (pt , qt ).
La stessa tesi vale se uno o entrambi dei qj è infinito, e T è limitato da Lpj in
L qj .
Applicando questo teorema con p0 = q0 = 1 e p1 = q1 = ∞, si ottiene immediatamente il seguente risultato.
Corollario 2.5. Se 1 < p ≤ ∞, M è limitato su Lp (X).
3. Calderón-Zygmund operators on spaces of homogeneous type
Gli operatori di Calderón-Zygmund (o a integrali singolari) costituiscono una
classe più ampia di quella degli operatori integrali. In questo paragrafo essi vengono presentati su generici spazi di natura omogenea; in questa generalità la loro
definizione risulta alquanto implicita, mentre nei contesti più comuni, come Rn o le
varietà differenziabili, la teoria delle distribuzioni ne consente una descrizione più
precisa.
Cominciamo con il considerare, dato uno spazio di natura omogenea X, un operatore lineare T con le seguenti proprietà:
(1) è definito sullo spazio Cc (X) delle funzioni continue a supporto compatto,
o su un suo sottospazio denso V ;
(2) è a valori in L1loc (X);
(3) esiste una funzione K(x, y), localmente integrabile su (X ×X)\{diagonale},
tale che, se f, g ∈ V e supp f ∩ supp g = ∅, allora
Z
ZZ
(3.1)
(T f )(x)g(x) dx =
K(x, y)f (y)g(x) dy dx ,
X
X×X
o, equivalentemente,
(3.2)
T f (x) =
Z
K(x, y)f (y) dy
X
per quasi ogni x 6∈ supp f .
Si noti che K non determina univocamente T : se, per esempio, T f = ϕf , le
tre proprietà sono soddifatte con K = 0, indipendentemente da ϕ. Infatti, il
Rnucleo K non descrive completamente l’operatore, perché non dice quanto valga
(T f )(x)g(x) dx se i supporti di f e g hanno intersezione non vuota.
X
Definizione. Si chiama operatore di Calderón-Zygmund un operatore lineare T
che
(i) sia limitato su Lq (X) per qualche q ∈ (1, ∞);
28
CHAPTER II
(ii) sia soddisfatta la condizione (3), ed esista una costante C > 0 tale che per
ogni coppia di punti distinti y, y 0 di X
Z
K(x, y) − K(x, y 0 ) dx < C ,
(3.3)
{x:d(x,y)>4cd(y,y 0 )}
dove c indica la costante nella disuguaglianza triangolare (2.1).
La condizione (3.3) si chiama condizione di Calderón-Zygmund6 .
La (3.3) va interpretata come una forma integrale di regolarità. Una condizione
di carattere puntuale che implica la (3.3) è la seguente: indichiamo con v(x, y) il volume della palla di centro x e raggio r = d(x, y), e supponiamo che per qualche α > 0
e per ogni terna di punti x, y, y 0 con d(x, y) > 4cd(y, y 0 ) valga la disuguaglianza
K(x, y) − K(x, y 0 ) ≤ C
(3.4)
d(y, y 0)α
,
v(x, y)d(x, y)α
(che appare come una condizione di Lipschitz7 in y, con una costante che dipende
dalla distanza da x).
Allora, posto Ej = {x : 4c2j d(y, y 0 ) ≤ d(x, y) < 4c2j+1 d(y, y 0)},
Z
K(x, y) − K(x, y 0 ) dx ≤
{x:d(x,y)>4cd(y,y 0 )}
Z
1
0 α
≤ Cd(y, y )
dx
α
{x:d(x,y)>4cd(y,y 0 )} v(x, y)d(x, y)
Z
∞
X
1
0 α
≤
Cd(y, y )
dx
α
Ej v(x, y)d(x, y)
j=0
Z
∞
X
1
0 −αj
dx .
≤
C2
Ej v(x, y)
j=0
Se x ∈ Ej , posto d = d(y, y 0) e r = d(x, y), allora B(x, r) ⊇ B(x, 4c2j d), per cui
v(x, y) ≥ m B(x, 4c2j d) . D’altra parte,
m(Ej ) = m B(x, 4c2j+1 d) − m B(x, 4c2j d) ≤ (c0 − 1)m B(x, 4c2j d) ,
per la (2.2). In conclusione,
Z
{x:d(x,y)>4cd(y,y 0 )}
∞
X
K(x, y) − K(x, y 0 ) dx ≤
C 00 2−αj ,
j=0
il che fornisce la condizione di Calderón-Zygmund.
Dimostreremo ora che gli operatori di Calderón-Zygmund sono di tipo debole
(1,1). La dimostrazione richiede due strumenti. Il primo è il lemma di ricoprimento
di Whitney per spazi di natura omogenea.
6 Il
coefficiente 4 nella (3.3) può essere sostituito da un qualunque numero maggiore di 1.
e nel seguito, diremo “lipschitziana di ordine α” in luogo della dizione più comune
“hölderiana di ordine α”.
7 Qui
29
Lemma 3.1. Sia F un chiuso non vuoto di X, e sia A il suo complementare.
Esistono costanti 1 < k < k 0 , indipendenti da F , e una famiglia numerabile di palle
Bj = B(xj , rj ) ⊂ A tali che
(i) le palle Bj sono a due a due disgiunte;
(ii) l’unione delle palle Bj∗ = B(xj , krj ) è uguale a A;
(iii) ogni palla Bj∗∗ = B(xj , k 0 rj ) ha intersezione non vuota con F .
Proof. Sia c la costante nella (2.1). Per ogni x ∈ A, sia dx = d(x, F ) e si prenda
Bx = B(x, δdx ), con δ < 1 da determinarsi. Si scelga quindi una famiglia {Bj =
Bxj }j∈J di tali palle che sia massimale rispetto alla proprietà di essere a due a due
disgiunte.
La famiglia {Bj } è numerabile. Ciò segue dal fatto che, fissati x0 ∈ X e un
intero n, le palle Bj di misura maggiore di 1/n e contenute in B(x0 , n) possono
essere solo in numero finito. Si noti anche che se una palla avesse misura nulla,
tutto X avrebbe misura nulla per la (2.2).
La (i) è dunque verificata. Per la (iii), basta prendere Bj∗∗ = B(xj , 2dxj ). Passiamo quindi alla (ii).
Si considerino le palle Bj∗ = B(xj , dxj /2). Esse sono chiaramente contenute in
A. Sia ora x ∈ A. Per la massimalità della famiglia {Bj }, la palla Bx interseca una
delle Bj . Vogliamo mostrare che x ∈ Bj∗ , se δ è stato scelto opportunamente. Sia
y un punto in Bx ∩ Bj e sia z ∈ F tale che d(xj , z) < 2dxj . Allora
dx ≤ d(x, z)
≤ c d(x, y) + d(y, z) ≤ c2 d(x, y) + d(y, xj ) + d(xj , z)
< c2 δdx + δdxj + 2dxj .
Quindi, se δc2 < 1,
dx <
(δ + 2)c2
dxj = σdxj .
1 − δc2
Ora
d(x, xj ) ≤ c d(x, y) + d(y, xj )
< cδ(dx + dxj )
< cδ(1 + σ)dxj .
Si tratta ora di prendere δ tale che
(
δ 1+
1
c2 (δ+2)c2
1−δc2
δ<
<
1
2
,
cioè δ < 1/(2 + 5c2 ). Il secondo risultato è la decomposizione di Calderón-Zygmund. Per semplicità
supponiamo che m(X) = ∞.
30
CHAPTER II
Lemma 3.2. Sia f ∈ L1 (X, m) e sia α > 0. È possibile decomporre f come
f (x) = g(x) +
∞
X
bj (x)
j=0
in modo che
(i) |g(x)| ≤ α;
(ii) le funzioni bj hanno supporto in palle Bj0 e sono tali che
1
m(Bj0 )
(iii)
P
j
m(Bj0 ) ≤
Z
Z
|bj (x)| dm(x) ≤ α ,
bj (x) dx = 0 ;
C
α kf k1 .
Proof. Sia A = {x : M f (x) > κα}, con κ da determinarsi, e si costruiscano le palle
Bj come nel Lemma 2.1. Si ponga quindi


[
B` 
Q0 = B0∗ \ 

`≥1
Q1 = B1∗ \ Q0 ∪
...

Qj = Bj∗ \ 
[
`<j
[
`≥2
Q` ∪

B` 
[
`>j

B`  .
I Qj danno una partizione di A e Bj ⊂ Qj ⊂ Bj∗ . Quindi
1
m(Qj )
Z
Qj
1
|f (x)| dm(x) ≤
m(Bj )
c0
≤
m(Bj∗ )
Z
Bj∗
|f (x)| dm(x)
Bj∗
|f (x)| dm(x)
Z
≤ c0 M1 f (xj )
≤ c0 κα ,
se c0 è la costante nella (2.2) e xj ∈ Bj∗ ⊂ A.
In particolare, se
Z
1
f (x) dm(x) ,
βj =
m(Qj ) Qj
risulta
|βj | ≤ c0 κα .
Si ponga allora
g = f χX\A +
X
j
βj χ Q j .
31
Poiché |f (x)| ≤ M f (x) quasi ovunque, risulta
|g(x)| ≤ c0 κα
quasi ovunque.
Si ponga ora
bj = (f − βj )χQj .
Allora bj ha supporto nella palla Bj∗ , ha integrale nullo e
1
m(Bj∗ )
Z
1
|bj (x)| dm(x) ≤
m(Bj∗ )
0
Z
Qj
|f (x)| dm(x) +
m(Qj )
|βj |
m(Bj∗ )
≤ (1 + c )κα .
Quindi se κ = 1/(1+c0 ) e Bj0 = Bj∗ , le condizioni (i) e (ii) sono verificate. Quanto
alla (iii), abbiamo
X
j
m(Bj∗ ) ≤ c0
X
j
m(Bj ) ≤ c0 m(A) ≤ C
kf k1
α
perché M è di tipo debole (1,1). Teorema 3.3. Un operatore T di Calderón-Zygmund è di tipo debole (1, 1).
Proof. Si supponga T limitato su Lq (X), e sia f ∈ L1 . Dato α > 0, si consideri la
decomposizione di Calderón-Zygmund
f (x) = g(x) +
∞
X
bj (x) ,
j=0
P
come dal Lemma 3.2, corrispondente al valore di α fissato. Se b(x) = j bj (x), si
ha
m {x : |T f (x)| > 2α} ≤ m {x : |T g(x)| > α} + m {x : |T b(x)| > α} .
Osserviamo ora che g ∈ Lq ; più precisamente, segue da (ii) e (iii) che
X
kbj k1 ≤ Ckf k1 ,
j
per cui anche kgk1 ≤ Ckf k1 . Usando quindi anche (i), si ha
kgkqq ≤ Cαq−1 kf k1
Poiché T è limitato su Lq (X) e per la disuguaglianza di Chebishev,
kT gkqq
m {x : |T g(x)| > α} ≤
αq
kgkqq
≤C q
α
kf k1
≤C
.
α
32
CHAPTER II
00
Passiamo ora a T b. Sia Bj0 = B(xj , rj ). Poniamo Bj = B(xj , 4crj ). Se x 6∈ Bj0 ,
si ha
Z
Z
T bj (x) =
K(x, y)bj (y) dy =
K(x, y) − K(x, xj ) bj (y) dy ,
Bj0
Bj0
in quanto bj ha integrale nullo. Allora
Z
00
X\Bj
|T bj (x)| dx ≤
=
≤
Z
Z
Z
00
X\Bj
Z
Bj0
Bj0
|bj (y)|
Bj0
|bj (y)|
≤C
Z
Bj0
K(x, y) − K(x, xj )|bj (y)| dy dx
Z
00
X\Bj
Z
K(x, y) − K(x, xj ) dx dy
{x:d(x,xj )>4cd(y,xj )}
|bj (y)| dy
K(x, y) − K(x, xj ) dx dy
≤ Cα|Bj0 | .
Di conseguenza
Z
X\
S
00
j Bj
|T b(x)| dx ≤ Cα
X
j
|Bj0 | ≤ Ckf k1 .
Per la disuguaglianza di Chebishev,
[ 00
kf k1
{x 6∈
.
Bj : |T b(x)| > α} ≤ C
α
j
Rimane da considerare la misura dell’insieme {x ∈
questa è sicuramente minore o uguale a
m
[
j
00
Bj
≤
X
j
00
m(Bj ) ≤ C
X
j
S
m(Bj0 ) ≤ C
00
j
Bj : |T b(x)| > α}. Ma
kf k1
.
α
Corollario 3.4. Un operatore di Calderón-Zygmund limitato su Lq (X) è anche
limitato su Lp (X) se 1 < p ≤ q. Se anche k ∗ (x, y) = k(y, x) soddisfa la condizione
di Calderón-Zygmund (3.3), allora T è limitato su Lp (X) per ogni p ∈ (1, ∞).
Proof. La prima parte dell’enunciato segue direttamente dal Teorema di interpolazione di Marcinkiewicz. Se q = ∞ non c’è altro da dimostrare.
0
Se q < ∞ e k ∗ soddisfa la (3.3), allora T ∗ , che è limitato su Lq (X), è pure un
operatore di Calderón-Zygmund, e dunque è limitato su Lr (X) per 1 < r ≤ q 0 . Ma
allora T è limitato su Lp (X) per q ≤ p < ∞. MIHLIN-HÖRMANDER MULTIPLIERS
33
4. Integral Lipschitz conditions
Lasciamo per questo paragrafo gli spazi di natura omogenea, e introduciamo
alcune nozioni preliminari al seguito del capitolo. Indicheremo con kxk la norma
euclidea di x ∈ Rn .
Si dice che una funzione f ∈ Lp (Rn ) soddisfa una condizione Lp -Lipschitz di
n
ordine α ∈ (0, 1), o che f ∈ Λα
p (R ), se esiste una costante c tale che
Z
(4.3)
p
|f (x − h) − f (x)| dx
1/p
≤ ckhkα
per ogni h ∈ Rn . Si pone in tal caso
kf kΛαp = kf kp + sup khk
(4.4)
−α
h6=0
Z
p
|f (x − h) − f (x)| dx
1/p
.
La norma (4.4) può essere sostituita da ciascuna delle norme equivalenti
kf k0Λαp
(4.5)
= kf kp +
khk
sup
0<khk<a
−α
Z
p
|f (x − h) − f (x)| dx
1/p
.
Infatti basta maggiorare, per khk ≥ a,
khk
Z
−α
p
|f (x − h) − f (x)| dx
1/p
≤ 2a−α kf kp .
Le condizioni di Lipschitz integrali sono localmente più deboli delle ordinarie
n
condizioni di Lipschitz. Per esempio, sia f (x) = kxk− p +α ϕ(x), dove 0 < α < 1 e
ϕ è una funzione C ∞ a supporto compatto uguale a 1 in un intorno di 0. Allora
Z
kx − hk− np +α ϕ(x − h) − kxk− np +α ϕ(x)p dx
≤
Z
kxk>2khk
+2
≤ Ckhk
= C|h|
Z
Z
Z
kx − hk− np +α ϕ(x − h) − kxk− np +α ϕ(x)p dx
kxk<3khk
kxk>2khk
kxk
kxk
−n+αp
r
= C|h|α .
n
Dunque f ∈ Λα
p (R ).
αp−p−1
dx
−n+αp−p
∞
2khk
1/p
dr
1/p
1/p
dx
+C
1/p
Z
+C
3khk
Z
r
0
1/p
3khk
r
0
αp−1
dr
αp−1
1/p
dr
1/p
34
CHAPTER II
α
Più in generale, lo spazio di Besov Bp,q
(Rn ) è definito, per α > 0 e 1 ≤ p, q ≤ ∞,
p
come lo spazio delle funzioni f ∈ L tali che
(
1/q
R
q dh
−α
se q < ∞
kf kp + Rn (khk kτh f − f kp ) khkn
α
kf kBp,q
=
−α
kf kp + suph6=0 khk kτh f − f kp
se q = ∞ ,
n
α
n
dove τh f (x) = f (x − h). Ovviamente Λα
p (R ) = Bp,∞ (R ).
n
Diamo ora due risultati riguardanti Λα
1 (R ) e che useremo nel prossimo paragrafo.
n
Lemma 4.1. Se f ∈ Λα
1 (R ), allora
|fˆ(ξ)| ≤ Ckf kΛα1 (1 + kξk)−α .
Proof. Si osservi che
Z
Z ξ
−ix·ξ
e
dx = − f (x)e−ix·ξ dx = −fˆ(ξ) .
f x−π
kξk2
Quindi
Z Z
ξ
1 −ix·ξ
−ix·ξ
ˆ
e
dx − f (x)e
dx
|f (ξ)| = f x − π
2
2
kξk
Z 1 ξ
dx
≤
−
f
(x)
f
x
−
π
2 kξk2
≤ Ckξk−α kf kΛα1 .
D’altra parte,
Quindi
|fˆ(ξ)| ≤ kf k1 ≤ kf kΛα1 .
(1 + kξk)α |fˆ(ξ)| ≤ C(1 + kξkα )|fˆ(ξ)| ≤ Ckf kΛα1 .
n
p
.
Lemma 4.2. Se f ∈ Λα
1 , allora f ∈ L per ogni p < n−α e kf kp ≤ Cp kf kΛα
1
R
Proof. Sia ϕ ∈ D(Rn ) tale che Rn ϕ = 1 e supp ϕ sia contenuto nella palla unitaria.
Poniamo
ψ0 (x) = ϕ(x) , ψj (x) = 2nj ϕ(2j x) − 2n(j−1) ϕ(2j−1 x) ,
cosı̀ che
nj
j
f = lim f ∗ 2 ϕ(2 ·) =
j→+∞
∞
X
j=0
f ∗ ψj .
Si ha kψj k1 ≤ 2 e kψj k∞ ≤ C2nj . Inoltre supp ψj ⊂ B(0, 2−(j−1) ) e
per j ≥ 1. Quindi, se j ≥ 1,
Z Z
dx
kf ∗ ψj k1 =
f
(x
−
y)
−
f
(x)
ψ
(y)
dy
j
n
Rn
R
Z Z
f (x − y) − f (x)|ψj (y)| dx dy
≤
Rn Rn
Z
α
≤ kf kΛ1
|y|α |ψj (y)| dy
Rn
≤ C2
−αj
kf kΛα1 .
R
Rn
ψj = 0
35
Inoltre kf ∗ ψj k∞ ≤ C2nj kf k1 ≤ C2nj kf kΛα1 . Quindi
Z
p
p−1
|f ∗ ψj (x)| dx ≤ C2nj(p−1)−αj kf kΛα1 .
kf ∗ ψj kp ≤ Ckf ∗ ψj k∞
Rn
Sommando su j, la serie delle norme di ordine p converge se p < n/(n − α). 5. Non-isotropic dilations in Rn and Calderón-Zygmund kernels
Per la parte restante di questo capitolo l’insieme ambiente X sarà Rn , dotato
della misura di Lebesgue. A completare la struttura di spazio di natura omogenea,
prenderemo in esame diverse quasi-distanze (tra cui quella euclidea), associate a
famiglie di dilatazioni.
Dati numeri positivi (non necessariamente interi) λ1 , . . . , λn , si chiamano dilatazioni non isotropiche di Rn relative agli esponenti λj le trasformazioni lineari
r · x = (r λ1 x1 , . . . , r λn xn ) .
Se Q = λ1 + · · · + λn , si ha chiaramente d(r · x) = r Q dx. Il numero Q si chiama
la dimensione omogenea di Rn rispetto alle date dilatazioni.
Si chiama norma omogenea associata alle date dilatazioni una funzione continua
x 7−→ |x| da Rn a [0, +∞) tale che
(1) |x| = 0 se e solo se x = 0;
(2) | − x| = |x| per ogni x;
(3) |r · x| = r|x|
Un esempio è dato da
|x| = |x1 |1/λ1 + · · · + |xn |1/λn .
La norma euclidea sarà indicata con k k.
Lemma 5.1. Sia | | una norma omogenea.
(i) Gli insiemi Br = {x : |x| ≤ r} sono compatti.
(ii) Esiste una costante c ≥ 1 tale che
|x + y| ≤ c |x| + |y|
per ogni x, y ∈ Rn .
(iii) Se λ0 = mini λi e λ00 = maxi λi , allora esistono due costanti A, B > 0 tali
che, per ogni x con |x| > 1,
0
00
A|x|λ ≤ kxk ≤ B|x|λ .
(iv) Esistono due costanti A0 , B 0 > 0 tali che, per ogni x con |x| < 1,
00
0
A0 |x|λ ≤ kxk ≤ B 0 |x|λ .
(v) Se | |0 è un’altra norma omogenea, allora | | e | |0 sono equivalenti, nel senso
che esistono costanti a, b > 0 tali che
a|x| ≤ |x|0 ≤ b|x|
per ogni x ∈ Rn .
36
CHAPTER II
Proof. Sia S la sfera unitaria chiusa nella norma euclidea, e sia m > 0 il minimo
della funzione | | su S. Dimostriamo che Bm è contenuto nella palla unitaria chiusa
euclidea B̃1 . Sia x tale che kxk > 1. Poiché limr→0 r · x = 0 e per la continuità
della norma omogenea, esiste δ < 1 tale che δ · x ∈ B̃1 e |δ · x| = δ|x| < m. Poiché
l’applicazione r 7−→ r · x è continua, esiste r ∈ [δ, 1) tale che r · x ∈ S. Allora
|r · x| ≥ m, da cui |x| = m/r > m.
Quindi Bm è limitato e dunque compatto. Ma allora Br = (r/m) · Bm è pure
compatto per ogni r > 0.
Sia c = max{|x + y| : x, y ∈ B1 }. Dati x, y 6= 0, sia t−1 = |x| + |y| > 0. Allora
t · x e t · x sono in B1 , per cui
t|x + y| = |t · (x + y)| = |t · x + t · y| ≤ c ,
e questo dimostra la (ii).
Per dimostrare la (iii), osserviamo che esistono s, σ > 0 tali che
B̃s ⊆ B1 ⊆ Bσ .
Quindi
r · B̃s ⊆ Br ⊆ r · B̃σ ,
per ogni r > 0. Se r > 1,
r · B̃s ⊇ B̃rλ0 s ,
r · Bσ ⊆ B̃rλ00 σ ,
e da questo segue facilmente la tesi. La (iv) si dimostra in modo analogo.
Per la (v), siano a, b rispettivamente il minimo e il massimo di |x|0 sulla sfera
S 0 = {x : |x| = 1}. Se x = r · y con y ∈ S 0 , allora
a|x| = ar ≤ r|y|0 = |x|0 ≤ rb = b|x| .
In particolare ogni norma omogenea è equivalente a
|x| =
n
X
j=1
|xj |1/λj .
Proposizione 5.2. Sia | | una norma omogenea associata a una famiglia di dilatazioni non isotropiche. Allora Rn , dotato della misura di Lebesgue e della quasidistanza d(x, y) = |x − y|, è uno spazio di natura omogenea.
Proof. Poiché la quasi-distanza è invariante per traslazioni, basta confrontare le
misure di Br e B2r . Ma m(Br ) = cr Q , dove c = m(B1 ), per cui la (2.2) vale con
c0 = 2 Q . Supporremo ora fissata una famiglia di dilatazioni non isotropiche e una corrispondente norma omogenea. Gli operatori che discuteremo saranno operatori di
convoluzione T f = f ∗ k, con k ∈ S 0 (Rn ).
37
La (3.1) equivale allora alla condizione che k coincida, su Rn \ {0}, con una
funzione k(x) ∈ L1loc (Rn \ {0}). In tal caso8
K(x, y) = k(x − y) .
La condizione di Calderón-Zygmund (3.3) equivale a richiedere che per ogni
h ∈ Rn , h 6= 0,
Z
k(x + h) − k(x) dx ≤ C .
(5.1)
|x|>4c|h|
Allo stesso modo, la (3.4) equivale a richiedere che k(x) sia lipschitziana9 di
ordine α > 0 fuori dall’origine e che esista C > 0 tale che per |x| > 4c|h| sia
(5.2)
α
k(x + h) − k(x) ≤ C |h|
.
|x|Q+α
Si noti che una distribuzione che soddisfi la condizione (ii) nella definizione di
operatore di Calderón-Zygmund non soddisfa necessariamente la condizione (i): si
prenda, per es., come k la funzione Ridenticamente uguale a 1. La (ii) è banalmente
verificata, ma T f (x) = f ∗ 1(x) = f non è limitato su nessun Lq . Per operatori
di convoluzione, l’ipotesi più naturale da imporre è la limitatezza su L2 (Rn ), che
equivale a richiedere che k̂ ∈ L∞ (Rn ).
Chiameremo quindi nucleo di Calderón-Zygmund una distribuzione k ∈ S 0 (Rn )
per cui valga la (5.1) e con k̂ ∈ L∞ (Rn ).
Vedremo ora un procedimento abbastanza generale di costruzione di nuclei di
Calderón-Zygmund. La distribuzione k si ottiene come “somma diadica”, a partire
da una successione di funzioni integrabili ϕj che, al variare di j ∈ Z, hanno norme
n
uniformemente limitate in qualche Λα
1 (R ), soddisfano una condizione di decadimento all’infinito, e hanno tutte media nulla. Ogni ϕj viene poi dilatata di un
fattore 2j .
Dati una funzione f e j ∈ Z, poniamo f (j) (x) = 2−Qj f (2−j · x).
Teorema 5.3. Sia {ϕj }j∈Z ⊂ L1 (Rn ) una famiglia di funzioni tali che esistano
costanti ε > 0, α ∈ (0, 1), C > 0 per cui valgano le seguenti proprietà:
R
(a) R |ϕj (x)|(1 + kxk)ε dx ≤ C;
(b) ϕj (x) dx = 0;
(c) kϕj kΛα1 ≤ C.
P
(j)
Allora la serie j∈Z ϕj converge in S 0 a un nucleo di Calderón-Zygmund.
Proof. Consideriamo la serie delle trasformate di Fourier
(5.3)
Xd
X
(j)
ϕj (ξ) =
ϕ
cj (2j · ξ) ,
j∈Z
8 Questa
j∈Z
notazione crea un’ambiguità tra la distribuzione k e la funzione k(x). Occorre non
confondere le due entità. Per esempio, quando si parla di trasformata di Fourier di k, questa va
intesa come distribuzione.
9 Nel senso della quasi-distanza d.
38
CHAPTER II
e dimostriamone la convergenza assoluta.
Se |2j · ξ| ≥ 1, usiamo il Lemma 4.1 e il Lemma 5.1 (iii) per ricavare che
−α
−αλ0
ϕ
.
cj (2j · ξ) ≤ C 1 + k2j · ξk
≤ C 0 1 + |2j · ξ|
Se invece |2j ·ξ| < 1, usando la (b), la disuguaglianza |eit −1| ≤ C|t|ε , e il Lemma
5.1 (iv), si ha
Z
j
ϕ
cj (2 · ξ) = −i(2j ·ξ)·x
− 1) dx
ϕj (x)(e
Z
j
ε
≤ Ck2 · ξk
|ϕj (x)|kxkε dx
Rn
Rn
0
≤ C 0 |2j · ξ|ελ .
Allora
X d
0
(j)
ϕj (ξ) ≤ C|ξ|−αλ
j∈Z
X
0
2−αλ j + C|ξ|ελ
u(ξ) =
X
j:2j |ξ|<1
j:2j |ξ|≥1
Poniamo dunque
0
0
2ελ j ≤ C 0 .
Xd
(j)
ϕj (ξ) ∈ L∞ (Rn ) .
j∈Z
Per convergenza dominata, data f ∈ S(Rn ),
Z
uf =
Rn
XZ
j∈Z
Rn
d
(j)
ϕj f ,
P
d
(j)
cioè u = j∈Z ϕj nel senso delle distribuzioni.
Se k = F −1 u, si ha dunque
X (j)
k=
ϕj
j∈Z
nel senso delle distribuzioni.
Rimane da dimostrare che, fuori dall’origine, k coincide con una funzione che
soddisfa la (5.1).
Mostriamo che la serie
X (j)
ϕj (x)
j∈Z
converge in L1 (K) per ogni compatto K ⊂ Rn \ {0}.
Possiamo supporre che K = {x : 2m ≤ |x| ≤ 2m+1 }, con m ∈ Z. Cambiando
variabile,
Z
2m ≤|x|≤2m+1
(j) ϕ (x) dx =
j
Z
2m−j ≤|x|≤2m+1−j
ϕj (x) dx .
39
Per j ≤ m, usiamo la (a) e il Lemma 5.1 (iii) per ottenere che
Z
2m−j ≤|x|≤2m+1−j
ϕj (x) dx ≤ 2(j−m)ελ0
≤ 2(j−m)ελ
0
Z
Z
2m−j ≤|x|≤2m+1−j
2m−j ≤|x|≤2m+1−j
0
≤ C2(j−m)ελ .
ϕj (x)|x|ελ0 dx
ϕj (x)kxkε dx
Per j > m, usiamo invece la (c) e il Lemma 4.2 per ottenere che
Z
2m−j ≤|x|≤2m+1−j
ϕj (x) dx ≤
Z
|x|≤2m+1−j
0
≤ C2(m−j)Q/p ,
ϕj (x)p dx
1/p
m(B2m+1−j )1/p
0
per un opportuno p > 1. Quindi
X
j∈Z
(j)
kϕj kL1 (K) ≤ C
X
0
2(j−m)ελ + C
X
0
2(m−j)Q/p ,
j>m
j≤m
dove entrambe le somme convergono.
Verifichiamo ora la condizione di Calderón-Zygmund. Sia h ∈ Rn \ {0}, e si
supponga 2m ≤ 4c|h| < 2m+1 . Allora
Z
|x|>4c|h|
X
k(x + h) − k(x) dx ≤
=
XZ
j∈Z
≤2
j∈Z
|y|>2m−j
XZ
j<m
≤C
≤C
X
|y|> 2
j<m
|x|>2m
(j)
ϕ (x + h) − ϕ(j) (x) dx
j
j
ϕj (y + 2−j · h) − ϕj (y) dy
m−j
2c
2(j−m)ελ
0
j<m
X
Z
0
XZ ϕj (y + 2−j · h) − ϕj (y) dy
|ϕj (y)| dy +
Z
j≥m
0
m−j
|y|> 2 2c
2(j−m)ελ + C
X
|ϕj (y)||y|ελ dy + C
0
2(m−j)αλ .
X
j≥m
k2−j · hkα
j≥m
Nel passaggio alla terza riga siè utilizzato il fatto che
2m−j 2m−j < |y| ≤ c |y + 2−j · h| + 2−j |h| < c |y + 2−j · h| +
,
2c
da cui segue che
|y + 2−j · h| >
2m−j
.
2c
40
CHAPTER II
6. Mihlin-Hörmander conditions on Fourier multipliers
Si consideri una famiglia di dilatazioni non isotropiche in Rn . La condizione di
Mihlin-Hörmander su un moltiplicatore di Fourier m(ξ) ha le seguenti caratteristiche:
(1) è invariante rispetto alle dilatazioni fissate, nel senso che se m(ξ) la soddisfa,
anche m(r · ξ) la soddisfa per ogni r > 0;
(2) implica che la distribuzione k = F −1 m è un nucleo di Calderón-Zygmund
adattato alle dilatazioni fissate.
Di conseguenza, l’operatore Sm f = m(i−1 ∂)f = f ∗ k è limitato su Lp (Rn ) per
ogni p ∈ (1, ∞).
Premettiamo la definizione e alcune proprietà degli spazi di Sobolev con esponente frazionario.
Definizione. Lo spazio di Sobolev H s (Rn ), con s ∈ R, consiste delle distribuzioni
f ∈ S 0 (Rn ) tali che fˆ è una funzione localmente integrabile e
Z
s
2
kf kH s =
|fˆ(τ )|2 1 + kτ k2 dτ < ∞ .
Rn
Vale l’inclusione H s (Rn ) ⊂ H t (Rn ) per s > t.
H (Rn ) = L2 (Rn ) per s > 0.
In particolare, H s (Rn ) ⊂
0
Lemma 6.1. Se s ∈ N, H s (Rn ) coincide con lo spazio delle funzioni f ∈ L2 (Rn )
tali che ∂ α f ∈ L2 (Rn ) per ogni multiindice α con |α| ≤ s.
Tralasciamo la dimostrazione, che segue facilmente dalla formula di Plancherel.
Lemma 6.2. Sia ϕ tale che ϕ̂ ∈ H s (Rn ) con s >
per ogni ε < s − n2
Z
Rn
|ϕ(x)| 1 + kxk
ε
n
2,
allora ϕ ∈ L1 (Rn ). Inoltre
dx ≤ Cε kϕ̂kH s .
Proof. Per la disuguaglianza di Hölder,
Z
Z
ε
|ϕ(x)|(1 + kxk) dx ≤
|ϕ(x)|(1 + kxk2 )ε/2 dx
Rn
≤C
Z
Rn
2
Rn
2 s
|ϕ(x)| (1 + kxk ) dx
≤ Cε kϕ̂kH s .
La conclusione segue dal fatto che
R
1/2 Z
Rn
1
dx
(1 + kxk2 )s−ε
(1 + kxk2 )−s+ε dx converge.
1/2
Si osservi che il Lemma 6.2 e il Teorema di Riemann-Lebesgue implicano la
immersione di Sobolev H s (Rn ) ⊆ C0 (Rn ) per s > n2 e la relativa disuguaglianza
(6.1)
kf k∞ ≤ Ckf kH s .
41
Lemma 6.3. Siano f ∈ H s (Rn ) e g ∈ S(Rn ). Allora f g ∈ H s (Rn ) e kf gkH s ≤
Cg kf kHs .
Proof. Poiché fcg = (2π)−n fˆ ∗ ĝ, si trova che
Z
Rn
|fcg(τ )|2 1 + kτ k2
s
−2n
Z
Z
n
2
s
0
0
0
ˆ
f (τ − τ )ĝ(τ ) dτ 1 + kτ k2 dτ
dτ = (2π)
R
Rn
Z Z
s
≤ Ckĝk1
|fˆ(τ − τ 0 )|2 |ĝ(τ 0 )| dτ 0 1 + kτ k2 dτ
n
n
Z
ZR R
s
0
|fˆ(τ )|2 1 + kτ + τ 0 k2 dτ dτ 0 .
|ĝ(τ )|
≤ Ckĝk1
Rn
Rn
Ma
per cui
1 + kτ + τ 0 k2 ≤ 1 + 2 kτ k2 + kτ 0 k2 ≤ 2 1 + kτ k2 1 + kτ 0 k2 ,
kf gk2H s
≤ Ckĝk1
Z
0
Rn
0 2
|ĝ(τ )| 1 + kτ k
dτ
0
Z
Rn
|fˆ(τ )|2 1 + kτ k2
s
dτ .
I due termini contenenti ĝ sono finiti, in quanto ĝ ∈ S(Rn ), e questo dimostra la
tesi. Siano ora a0 < a1 < b1 < b0 numeri positivi. Indichiamo con η una funzione in
D(Rn ) tale che10
(i) supp η ⊆ {ξ : a0 ≤ |ξ| ≤ b0 },
(ii) η(ξ) ≥ 0 per ogni ξ e η(ξ) > 0 per a1 ≤ |ξ| ≤ b1 .
Poniamo inoltre mr (ξ) = m(r · ξ).
Definizione. Si chiama moltiplicatore di Mihlin-Hörmander, adattato alle dilatazioni fissate, una funzione m(ξ) tale che
sup kmr ηkH s = kmkM Hs < ∞
r>0
per qualche s >
n
2.
Indicheremo con M Hs (Rn ) la classe dei moltiplicatori su Rn con kmkM Hs < ∞.
Si noti che la definizione stessa implica che m e mr hanno la stessa norma M Hs
per ogni r > 0. Inoltre, per la (6.1), le norme kmr ηk∞ sono uniformemente limitate,
da cui segue che m ∈ L∞ (Rn ).
Per s intero, la condizione puntuale
(6.2)
P
α
∂ m(ξ) ≤ C|ξ|− λi αi
per ξ 6= 0 e |α| ≤ s implica che m ∈ M Hs (Rn ). Questa è la condizione inizialmente
enunciata da Mihlin (per dilatazioni isotropiche).
10 Si
noti che nelle condizioni (i) e (ii) si può sostituire la norma euclidea alla norma omogenea.
42
CHAPTER II
Si può dimostrare, usando il Lemma 6.3, che scelte diverse di η inducono norme
M Hs equivalenti, per cui la condizione di Mihlin-Hörmander non dipende dalla
scelta di η. Per i nostri scopi è utile scegliere η in modo che
X
(6.2)
j∈Z
η(2j · ξ) = 1
per ogni ξ 6= 0. Per ottenere una tale η, si parta da una η0 ∈ D(Rn ) soddisfacente
(i) e (ii) con a1 ≤ 1 e b1 ≥ 2. Posto
η̃(ξ) =
X
j∈Z
η0 (2j · ξ) ,
la funzione η = η0 /η̃ soddisfa (i), (ii) e la (6.2).
Teorema 6.4. Sia m un moltiplicatore di Mihlin-Hörmander, adattato a una fissata famiglia di dilatazioni. Allora l’operatore Sm = m(i−1 ∂) è limitato su Lp (Rn )
per 1 < p < ∞.
Proof. Sia mj (ξ) = m(2−j · ξ)η(ξ). Allora
X
j∈Z
mj (2j · ξ) = m(ξ)
quasi ovunque e nel senso delle distribuzioni. Ne segue che, se poniamo
k = F −1 m , ϕj = F −1 mj ,
allora
k=
X
(j)
ϕj
j∈Z
nel senso delle distribuzioni. Mostriamo ora che le ϕj soddisfano le ipotesi (a), (b),
(c) del Teorema 5.3.
Poiché kmj kH s ≤ kmkM Hs , la (a) segue dal Lemma 6.2. La (b) segue dal fatto
che mj (0) = 0.
Dimostriamo ora la (c) con α = 1, osservando preliminarmente che per ogni
i = 1, . . . , n e per ogni j ∈ Z, ξi mj (ξ) ∈ H s (Rn ) e che kξi mj kH s ≤ CkmkM Hs .
Infatti, sia ω(ξ) ∈ D(Rn ), con ω = 1 sul supporto di η. Allora ξi mj = (ξi ω)mj , e
possiamo dunque applicare il Lemma 6.3.
Usando la disuguaglianza di Hölder come nella dimostrazione del Lemma 6.2, si
ottiene che
Z
Rn
ϕj (x − h) − ϕj (x) dx ≤ C
Z
Rn
ϕj (x − h) − ϕj (x)2 1 + kxk2 s dx
21
.
Essendo mj a supporto compatto, ϕj è C ∞ , per cui
Z
2 ϕj (x − h) − ϕj (x) = 1
0
2
Z
2
h · ∇ϕj (x − th) dt ≤ khk
1
0
k∇ϕj (x − th)k2 dt .
43
Quindi, poiché ∂\
xi ϕj = iξi mj ,
Z
−1
ϕj (x − h) − ϕj (x) dx ≤
sup khk
0<khk<1
Rn
≤
≤
≤
sup
0<khk<1
sup
0<khk<1
sup
0<khk<1
≤2
=2
Z
Rn
X
n
i=1
khk
Z
Z
−1
Rn
1
0
Z
Z
Z
Rn
1
ϕj (x − h) − ϕj (x)2 1 + kxk2 s dx
2
k∇ϕj (x − th)k dt 1 + kxk
0
2
Rn
k∇ϕj (x)k 1 + kx + thk
2
k∇ϕj (x)k 1 + kxk
kξi mj k2H s
21
2 s
dx
21
2 s
2 s
dx
dx dt
21
12
21
≤ CkmkM Hs .
Usando la norma (4.5) e il fatto che kϕj k1 ≤ CkmkM Hs per il Lemma 6.2, si ha
che kϕj kΛ11 ≤ CkmkM Hs .
Applicando allora il Teorema 5.3 si ha la conclusione. 7. Applications
Results about Lp -boundedness for Fourier multipliers have important consequences for differential operator with constant coefficients.
Our first application concerns spectral multipliers. We shall make the following
two assumptions on the symbol P (ξ) of the operator L = P (i−1 ∂):
(1) there are dilations r · ξ = (r λ1 ξ1 , . . . , r λn ξn ) such that P is homogeneous of
degree k > 0, i.e. P (r · ξ) = r k P (ξ);
(2) P (ξ) > 0 for ξ 6= 0.
Condition (1) is equivalent to saying that each α such that the monomial ξ α has
a non-zero coefficient in P has a non-isotropic degree
d(α) =
n
X
λj α j
j=1
equal to k.
Condition (2) implies that L is self-adjoint and positive with domain D = {f ∈
L2 : Lf ∈ L2 }.
Theorem 7.1. Let L = P (i−1 ∂) with P satisfying (1) and (2), and let m(λ) ∈
M Hs (R) with s > n2 . Then m(L) is bounded on Lp (Rn ) for 1 < p < ∞.
Observe that the Mihlin-Hörmander condition on the real line makes sense also
for multipliers defined only on R+ or on R− . Moreover, a multiplier m on the whole
44
CHAPTER II
line satisfies the Mihlin-Hörmander condition if and only if both m± = mχR± do.
Since the spectrum of L is the positive half line, it is sufficient to assume that m is
defined for λ > 0.
The proof of Theorem 7.1 requires some further remarks on Sobolev spaces and
non-isotropic norms.
Lemma 7.2. Let T be a linear operator, bounded from H s0 (Rn ) to H s0 (Rm ) and
from H s1 (Rn ) to H s1 (Rm ), with 0 ≤ s0 < s1 , and let C0 , C1 be the corresponding
operator norms. Then, for s0 < s < s1 , T is bounded from H s (Rn ) to H s (Rm ). If
s = (1 − θ)s0 + θs1 , then the operator norm of T acting between the H s spaces is
not larger than C01−θ C1θ .
Proof. Consider T 0 = F T F −1 . By assumption, for j= 0, 1, T 0 is bounded from
the weighted L2 spaces L2sj = L2 Rn , (1 + kτ k2 )sj dτ to the same space on Rm ,
and we want to prove that it is bounded between the L2s spaces, with the stated
bound on the norm.
Like in the proof of the Riesz-Thorin theorem, we use the three-lines theorem.
It is sufficient to prove to prove that, if g, h are continuous functions with compact
support in Rn and Rm respectively, and kgkL2s = khkL2s = 1, then
Z
(7.1)
0
Rm
(T g)(x)h(x) 1 + kxk
For z ∈ C, define
gz (x) = g(x) 1 + kxk2
and let
F (z) =
Z
Rm
z
,
2 s
dx ≤ C01−θ C1θ .
hz (x) = h(x) 1 + kxk2
(T 0 gz )(x)hz (x) 1 + kxk2
s+2z
z
,
dx .
Since gz , hz have compact support, F is defined and holomorphic in the whole
plane. We restrict F to the vertical strip S where
s1 − s
s0 − s
≤ <ez ≤
.
2
2
For z ∈ S,
|F (z)| ≤
Z
Rm
0
(T gz )(x)hz (x) 1 + kxk2 s1 dx
≤ C1 kgz kL2s khz kL2s .
1
1
If g and h are supported on the ball of radius r, using the normalization of g
and h in L2s ,
Z
2s1 −s
2
kgz kL2s =
|g(x)|2 1 + kxk2
dx
1
kxk<r
≤ (1 + r 2 )2(s1 −s) ,
and similarly for hz . Hence F is bounded on S.
45
sj −s
2 ,
For <ez = σj =
|F (z)| ≤ Cj kgz kL2σ khz kL2σ
j
j
= Cj kgkL2s khkL2s
= Cj .
By the three lines theorem, if <ez = (1 − θ)σ0 + θσ1 = 0,
|F (z)| ≤ C01−θ C1θ .
For z = 0 this gives (7.1). Lemma 7.3. Given a family of dilations on Rn , with homogeneous dimension Q.
Let hxi be a continuous function from Rn to R, homogeneous of degree 1 with respect
to the given dilations, and strictly positive for x 6= 0 (e.g. a homogeneous norm 11 ).
Let S be the set where hxi = 1. There is a positive Borel measure σ on S such that
Z
Z ∞Z
f (x) dx =
f (r · x) dσ(x) r Q−1 dr ,
Rn
0
S
for every integrable function f .
Proof. If E is a Borel subset of S, let
E ] = {r · x : x ∈ E , r ≤ 1} ,
and define
σ(E) = Qm(E ] ) .
For 0 < a < b, let
Ea,b = {r · x : x ∈ E , a < r ≤ b} = (b · E ] ) \ (a · E ] ) .
Then
bQ − a Q
σ(E) =
m(Ea,b ) =
Q
Z
r Q−1 dr dσ .
E×[a,b]
Standard measure-theoretic arguments give the conclusion.
Proposition 7.4. Let f ∈ H s (R), s ≥ 0, be supported on interval [b, 2b], with
b > 0. If P satisfies (1), (2), then f ◦ P ∈ H s (Rn ), and
kf ◦ P kH s (Rn ) ≤ C(b, P )kf kH s (R) .
Proof. If s = m ∈ N, we use the characterization of H m as the space of L2 functions
with derivatives in L2 up to order m.
1
Set hξi = P (ξ) k . By Lemma 7.3,
kf ◦
P k22
11 The
= σ(S)
Z
(2b)1/k
b1/k
k 2 Q−1
f (r ) r
dr = C
missing hypothesis is that h−xi = hxi.
Z
2b
b
f (r)2 r Qk −1 dr ≤ Cb Qk −1 kf k2 .
2
46
CHAPTER II
Similar estimates hold for the L2 norms of ∂ α (f ◦ P ), with |α| ≤ m, by the chain
rule and Leibniz’s rule.
Assume now that m < s < m + 1. Let ω be a smooth function on the positive
half-line, equal to 1 on [b, 2b] and with compact support. Define the operator
T g(ξ) = (gω) P (ξ) ,
mapping functions on R to functions on Rn . By the first part of the proof and
Lemma 6.3,
kT gkH m (Rn ) ≤ C(b, P )kgωkH m (R) ≤ C(b, P, ω)kgkH m(R) ,
and similarly for H m+1 . By Lemma 7.2,
kT gkH s (Rn ) ≤ C(b, P, ω)kgkH s(R) .
If supp f ⊆ [b, 2b], then T f = f ◦ P and does not depend on the choice of ω. We can prove now Theorem 7.1.
Proof of Theorem 7.1. Let m̃(ξ) = m P (ξ) . If η satisfies (i) and (ii) of Section 6
on R+ , then η ◦ P satisfies the same conditions on Rn . Since
m̃r (ξ) = m P (r −1 · ξ) = m r −1 P (ξ) = mr P (ξ) ,
we have that
km̃r (η ◦ P )kH s (Rn ) = k(mr η) ◦ P kH s (Rn ) ≤ Ckmr ηkH s (R) .
Therefore m̃ ∈ M Hs (Rn ). Here is a corollary which shows the importance of looking for minimal assumptions on the multipliers (in terms of the Sobolev spaces they must belong to).
Corollary 7.5. Let γ ∈ R and 1 0,
(n+ε) 21 − p1 iγ
.
kL kLp →Lp ≤ Cp,ε 1 + |γ|
Proof. We apply Theorem 7.1 to m(λ) = λiγ . If η ∈ D(R+ ) and r > 0,
kmr ηkH s = kmηkH s .
If s = k ∈ N, kmηkH k ≤ Ck (1 + |γ|)k , by estimating L2 -norms of derivatives.
Assume now that k < s < k + 1, s = θk + (1 − θ)(k + 1). Setting u = mη
c and using
Hölder’s inequality,
Z
2
kmηkH s =
|u(τ )|2 (1 + |τ |2 )s dτ
R
≤
Z
2
R
2 k
|u(τ )| (1 + |τ | ) dτ
2(1−θ)
= kmηk2θ
H k kmηkH k+1
2s
≤ Cs 1 + |γ|
.
θ Z
2
R
2 k+1
|u(τ )| (1 + |τ | )
dτ
1−θ
47
Then kmkM Hs ≤ Cs (1 + |γ|)s . By Theorem 7.1, kLiγ kLp →Lp ≤ C(1 + |γ|)s for
every p ∈ (1, +∞). This is not yet the required estimate, but we shall use this
partial result to complete the proof.
It is sufficient to take p > 2, by duality. The Plancherel formula gives that
iγ
kL kL2 →L2 = 1. Take p0 > p, and let θ ∈ (0, 1) be such that
1
1
θ
1−θ
1
1
.
=
+
= −θ
−
p
p0
2
2
2 p0
By the Riesz interpolation theorem,
kLiγ kLp →Lp ≤ Cp (1 + |γ|)sθ = Cp (1 + |γ|)
s
1−1
2
p
1− 1
2
p0
.
If we let p0 tend to ∞, the exponent decreases to 2s 21 − p1 , and if s tends to
n/2, this quantity dcereases to n 12 − p1 . Given ε > 0, it is then possible to find s
and p0 such that the required estimates holds. Our second application concerns a-priori estimates. We keep condition (1) at
the beginning of this Section, and replace (2) by
(2’) P (ξ) 6= 0 for ξ 6= 0.
Theorem 7.6. Assume that P satisfies (1) and (20 ), and let α = (α1 , . . . , αn ) be
a multi-index with d(α) ≤ k. Then for every f ∈ S(Rn ) and 1 < p < ∞,
k∂ α f kp ≤ Cp kf kp + kLf kp .
Proof. Take ϕ ∈ D(Rn ) such that ϕ(ξ) = 1 on some neighborhood of 0. Then
α
(iξ)
1
−
ϕ(ξ)
α f (ξ) = (iξ)α ϕ(ξ)f(ξ)
ˆ +
c (ξ)
∂d
Lf
P (ξ)
ˆ + m2 (ξ)Lf(ξ)
c
= m1 (ξ)f(ξ)
.
The multiplier m1 is in D(Rn ), so that u = F −1 m1 ∈ S(Rn ) and
−1
F (m1 fˆ) ≤ kf ∗ ukp ≤ kuk1 kf kp .
p
We verify now that m2 satisfies (6.2) for arbitrary multi-indices β. Because m2
is smooth, these estimates are trivial when ξ is in the support of ϕ. We can then
restrict ourselves to ξ large enough so that ϕ(ξ) = 0.
In this region, m2 is homogeneous of degree −k + d(α) ≤ 0, hence it is bounded.
Any derivative ∂ β m2 is homogeneous of degree −k + d(α) − d(β) ≤ −d(β). Hence
It follows that
β
∂ m2 (ξ) ≤ Cβ |ξ|−d(β) .
−1
F (m2 fˆ) ≤ Cp kf kp .
p
48
CHAPTER II
MARCINKIEWICZ MULTIPLIERS
49
CHAPTER III
LITTLEWOOD-PALEY THEORY
AND MARCINKIEWICZ MULTIPLIERS
1. Square functions
Sia I = [0, 1]. La funzione di Rademacher rn ∈ L2 (I) è definita, per n ≥ 0, da
rn (t) = (−1)[2
n
t]
.
In altri termini, decomponendo I nell’unione degli intervalli
[j2−n , (j + 1)2−n ] ,
j = 0, . . . , 2n − 1 ,
rn assume il valore costante (−1)j su ciascun intervallo.
Lemma 1.1. Se k ≥ 1 e 0 < n1 < n2 < · · · < nk , allora
Z
1
0
rn1 (t)rn2 (t) · · · rnk (t) dt = 0 ,
e le funzioni di Rademacher formano un sistema ortonormale, ma non completo,
in L2 (I).
Proof. Il primo asserto è ovvio per k = 1. Supponiamo dunque k ≥ 2. Su ognuno degli intervalli [j2−nk−1 , (j + 1)2−nk−1 ] il prodotto rn1 (t)rn2 (t) · · · rnk−1 (t) è
costante, mentre rnk (t) assume i valori ±1 su sottoinsiemi di uguale misura. Quindi
l’integrale dell’intero prodotto è nullo su ciascuno di tali intervalli.
L’ortonormalità è dunque ovvia. Si osservi infine che la funzione r1 r2 è ortogonale
a tutte le rn per concludere che il sistema non è completo. La rilevanza delle funzioni di Rademacher è dovuta al seguente risultato, noto
come teorema di Khintchin.
P∞
Teorema 1.2. Sia f (t) = n=0 an rn (t) ∈ L2 (I). Allora per ogni p < ∞, la norma
di f in Lp è equivalente alla norma di f in L2 , ossia
cp
X
∞
n=0
|an |
2
1/2
≤ kf kp ≤ Cp
X
∞
n=0
|an |
2
1/2
.
Typeset by AMS-TEX
50
CHAPTER III
Proof. Supponiamo inizialmente p > 2. Per la disuguaglianza di Hölder, kf k2 ≤
kf kp . Per dimostrare la disuguaglianza opposta, è sufficiente prendere p = 2k ed f
reale. Si ha
2k
Z 1X
∞
2k
dt
an rn (t)
kf k2k =
0
(1.1)
=
n=0
X
(n1 ,...,n2k )∈N2k
Z
1
an1 an2 · · · an2k rn1 (t)rn2 (t) · · · rn2k (t) dt .
0
Per il Lemma 1.1, gli addendi non nulli nella (1.1) possono essere solo quelli in
cui uno stesso indice compare un numero pari di volte. In tal caso l’integrando è
una costante. Pertanto
X
kf k2k
≤
C
a2n1 a2n2 · · · a2nk ,
k
2k
n1 ≤···≤nk
dove Ck è un maggiorante del numero di elementi di N2k in cui compaiono ripetuti
due volte gli indici n1 ≤ · · · ≤ nk .
Ma allora
X
kf k2k
≤
C
a2n1 a2n2 · · · a2nk
k
2k
(n1 ,...,nk )∈Nk
= Ck
X
∞
n=0
a2n
k
,
che fornisce la tesi per p > 2.
Se 1 < p < 2, dalla disuguaglianza di Hölder segue che kf kp ≤ kf k2 . Sempre
per la disuguaglianza di Hölder, e per la parte precedente della dimostrazione,
kf k22 ≤ kf kp kf kp0 ≤ Cp0 kf kp kf k2 .
Quindi kf k2 ≤ Cp0 kf kp .
Rimane da considerare il caso p = 1. Procedendo come sopra,
kf k24/3 ≤ kf k1 kf k2 ≤ Ckf k1 kf k4/3 ,
da cui kf k4/3 ≤ Ckf k1 . Essendo anche kf k1 ≤ kf k4/3 per la disuguaglianza di
Hölder, la dimostrazione è completata. Corollario 1.3. Sia Tn una successioni di operatori limitati su Lp (X), dove X è
uno spazio di misura e p < ∞. Se esiste una costante A tale che per ogni scelta
possibile dei segni εn = ±1, risulta
k
(1.2)
∞
X
n=0
εn Tn kpp ≤ A ,
allora vale la maggiorazione
∞
1/2 X
2
≤ Cp Akf kp .
|T
f
|
n
n=0
p
51
Proof. Preso t ∈ [0, 1], si consideri l’operatore
Tt =
∞
X
rn (t)Tn ,
n=0
dove rn è l’n-esima funzione di Rademacher. Per ipotesi,
kTt f kpp ≤ Ap kf kpp .
Allora anche
Z
1
0
Z
p
X
|Tt f (x)| dx dt =
Z
1
0
kTt f kpp dt ≤ Ap kf kpp .
Cambiando ordine di integrazione, si ha internamente
Z
1
p
0
|Tt f (x)| dt =
Z
1
0
≥ cp
p
∞
X
dt
r
(t)T
f
(x)
n
n
n=0
X
∞
n=0
|Tn f (x)|
2
p/2
per il Teorema 1.2. Quindi
Z X
∞
X
n=0
|Tn f (x)|
2
p/2
dx ≤
≤
c−1
p
Z Z
X
1
0
p
p
c−1
p A kf kp
|Tt f (x)|p dt dx
,
come da dimostrarsi. Il Corollario 1.3 può essere visto nel modo seguente. Si consideri lo spazio L p (`2 )
costituito dalle successioni F = {fn } di funzioni misurabili su X tali che F (x) =
{fn (x)} ∈ `2 per quasi ogni x ∈ X e inoltre
kF kLp (`2 ) =
Z
X
kF (x)kp`2
dx
1/p
<∞.
Il Corollario 1.3 afferma che, sotto l’ipotesi (1.2), l’operatore
Tf = {Tn f }
è limitato da Lp a Lp (`2 ). Per dualità si ha allora il seguente corollario.
Corollario 1.4. Sia Tn una successione di operatori che soddisfino la (1.2), e sia
1 < p < ∞. Allora, data F = {fn } ∈ Lp (`2 ), risulta
∞
X
≤ Cp AkF kLp (`2 ) .
T
f
n
n
n=0
p
52
CHAPTER III
Proof. La (1.2) implica la stessa maggiorazione per gli operatori Tn∗ e con p0 al
posto di p. Essendo p > 1, p0 è finito. Dunque l’operatore Uf = {Tn∗ f } è limitato
0
0
0
da Lp a Lp (`2 ). Di conseguenza, U∗ è limitato dal duale di Lp (`2 ) a Lp .
Data G ∈ Lp (`2 ), si ponga
hF |Gi =
Z X
∞
fn (x)gn (x) dx .
X n=0
Si verifica facilmente che le applicazioni lineari F →
7 hF, Gi sono tutti e soli
0
p0 2
i funzionali continui su L (` ), ossia lo spazio duale di Lp (`2 ) si identifica con
Lp (`2 ). Inoltre
hU∗ F |gi = hF |Ugi
∞ Z
X
=
fn (x)Tn∗ g(x) dx
=
n=0 X
∞ Z
X
n=0
=
Z X
∞
X
Quindi U∗ F =
P∞
n=0
Tn fn (x)g(x) dx
X
n=0
Tn fn g(x) dx .
Tn fn , da cui la segue la tesi. 2. Littlewood-Paley functions
Combining the results in the previous Section with the Calderón-Zygmund theory, we shall obtain the basic properties of Littlewood-Paley functions.
on Rn we consider a family of dilations x 7−→ r · rx, and call Q the resulting
homogeneous dimension of Rn . If f is defined on Rn , we set f (j) (x) = 2−Qj f (2−j ·x).
We shall take at various stages functions ψ ∈ S(Rn ) satisfying
(2.1)
0 ≤ ψ̂ ∈ D(Rn ) and 0 6∈ supp ψ̂ ;
sometimes we shall also impose one of the two following conditions:
(2.2)
X
j∈Z
(2.2’)
X
j∈Z
ψ̂(2j · ξ) > 0 , for ξ 6= 0 ,
ψ̂(2j · ξ) = 1 , for ξ 6= 0 ,
Observe that condition (2.2) can be obtained by imposing that ψ̂(ξ) > 0 for
1 ≤ |ξ| ≤ 2, if | | is a homogeneous norm for the given dilations.
There are different ways to obtain a ψ satisfying (2.1) and (2.2’). Starting with ψ 0
satisfying (2.1) and (2.2), and denoting by s(ξ) the sum in (2.2), set ψ = F −1 (ψ̂0 /s).
Another way is to take ϕ ∈ D(Rn ) so that ϕ(ξ) = 1 on a neighborhood of 0, and set
53
ψ̂(ξ) = ϕ(ξ) − ϕ(2 · ξ). We remark that, conversely, if ψ satisfies (2.1) and (2.2’),
the function
X

ψ̂(2j · ξ) if ξ 6= 0 ,
(2.3)
ϕ(ξ) = j≥0

1
if ξ = 0 ,
is in D(Rn ), ϕ(ξ) = 1 on a neighborhood of 0, and ψ̂(ξ) = ϕ(ξ) − ϕ(2 · ξ).
Proposition 2.1.PSuppose ψ satisfies (2.1) and (2.20 ). If f ∈ Lp (Rn ), and 1 <
p < ∞, the series j∈Z f ∗ ψ (j) converges to f in Lp .
Proof. Let ϕ be the function in (2.3) and let u = F −1 ϕ. The u(j) form an approximate identity for j → −∞, so that
lim f ∗ u(j) = f
j→−∞
in Lp . We prove now that
lim kf ∗ u(j) kp = 0 .
j→∞
For f continuous with compact support, this follows from
0
kf ∗ u(j) kp ≤ kf k1 ku(j) kp ≤ C2−Qj/p .
For a general f ∈ Lp , given δ > 0, take g continuous with compact support such
that kf − gkp < δ. If j is large enough, kg ∗ u(j) kp < δ, so that
kf ∗ u(j) kp ≤ k(f − g) ∗ u(j) kp + kg ∗ u(j) kp
≤ kf − gkp ku(j) k1 + kg ∗ u(j) kp
< 2δ .
Therefore
N
X
j=−M
f ∗ψ
(j)
=
N
X
j=−M
f ∗ (u(j) − u(j+1) )
= f ∗ u(−M ) − f ∗ u(N +1) ,
and the conclusion follows.
If {fj }j∈Z is a sequence of Lp -functions on Rn , we set
{fj }
Lp (`2 )
i.e. the norm in Lp Rn , `2 (Z) .
=
Z
Rn
X
j∈Z
|fj (x)|
2
p2
dx
p1
,
54
CHAPTER III
Teorema 2.2. Assume that ψ ∈ S(Rn ) satisfies (2.1). Then, if 1 < p < ∞ and
f ∈ Lp (Rn ),
k{f ∗ ψ (j) }kLp (`2 ) ≤ C(ψ, p)kf kp .
(2.4)
If, in addition, ψ also satisfies (2.2), then
kf kp ∼ k{f ∗ ψ (j) }kLp (`2 ) ,
(2.5)
i.e. the two norms are equivalent for f ∈ Lp (Rn ). In other words, defining the
Littlewood-Paley function
Sf (x) =
X
j∈Z
we have kSf kp ∼ kf kp .
f ∗ ψ (j) (x)2
12
,
Proof. We apply Corollary 1.3 to the operators Tj f = f ∗ ψ (j) . By Theorem 5.3 of
P
Chapter II, for every choice of the signs εj , the series j∈Z εj ψ (j) converges to a
Calderón-Zygmund kernel, provided ψ satisfies (2.1). Since the constants appearing
in
P the estimates do not depend on the choice of signs,, the operator norms of the
j∈Z εj Tj are uniformly bounded. This gives (2.4).
Assume now that ψ also satisfies (2.2) and consider the function
(2.6)
b(ξ) =
X
j∈Z
d
(j) (ξ)2 =
ψ
X
j∈Z
ψ̂(2j · ξ)2 ,
By (2.1), the support of ψ̂ is contained in a rim 0 < a ≤ |ξ| ≤ b, so that for every
ξ 6= 0 at most N ∼ log2 (b/a) terms in the series (2.7) are different from 0. This
implies that b(ξ) is smooth for ξ 6= 0. Since b(2 · ξ) = b(ξ),
inf b(ξ) = min b(ξ) > 0 .
ξ6=0
1≤|ξ|≤2
Let η ∈ S(Rn ) be the function such that
η̂(ξ) =
ψ̂(ξ)
.
b(ξ)
Then η too satisfies (2.1), so that, arguing as before, the operators Tj0 f = f ∗ η (j)
satisfy (1.2). By Corollary 1.4,
X
(j)
(j) f ∗ ψ ∗ η ≤ Ck{f ∗ ψ (j) }kLp (`2 ) .
p
j∈Z
Consider finally ψ ∗ η. Since
X
j∈Z
[
ψ
∗ η(2j · ξ) =
X ψ̂(2j · ξ)2
=1,
b(2j · ξ)
j∈Z
55
it satisfies (2.2’) together with (2.1), so that
X
X
f ∗ (ψ ∗ η)(j) =
f ∗ ψ (j) ∗ η (j) = f .
j∈Z
j∈Z
This proves (2.5). We shall prove below a multi-parameter version of Theorem 2.2. But before
that, we present the general aspects of the “multi-parameter theory”.
The Calderón-Zygmund theory is often referred to as the one-parameter singular
integral theory, because the assumptions made on the kernels are adapted to a given
family of dilations depending on one parameter r > 0. From the point of view of
Fourier multipliers, the same can be said for the Mihlin-Hörmander condition.
In the multi-parameter theory (also called product theory) one has a finite family
of spaces Rni , each with its own dilations xi 7−→ r · xi , and on the product RN =
Rn1 × · · · × Rnk one defines
(2.7)
r(x1 , . . . , xk ) = (r1 · x1 , . . . , rk · xk ) ,
for r = (r1 , . . . , rk ) ∈ (R+ )k .
The simplest example of an operator arising in the multi-parameter theory is the
convolution operator T f = f ∗ K, where K is the tensor product
(2.8)
K(x) = K1 (x1 ) · · · Kk (xk )
of Calderón-Zygmund kernels on the various Rni . Since each Ki is singular (i.e.
non-locally integrable) only at the origin, the product kernel K is singular on the
union of the “coordinate subspaces” xi = 0.
However, we shall not discuss product kernels, but we shall instead restrict ourselves to the Fourier multipliers connected with the product theory, the Marcinkiewicz multipliers. The simplest example is the product
(2.9)
m(ξ) = m1 (ξ1 ) · · · mk (ξk )
of Mihlin-Hörmander multipliers on the Rni .
The general product theory is however less trivial than what these examples
maight suggest12 . The proofs are based on the one-parameter theory and on Littlewood-Paley decompositions.
On each Rni fix a ψi satisfying (2.1), and, for J = (j1 , . . . , jk ) ∈ Zk , let
(j )
(j )
ψ (J ) (x) = ψ1 1 (x1 ) · · · ψk k (xk ) .
We then construct
X
1
2 2
(J )
Sf (x) =
f ∗ ψ (x)
,
J ∈Zk
12 In
fact, there is no difficulty in proving that operators f 7−→ f ∗ K, with K as in (2.8), or
f 7−→ F −1 (mfˆ), with m as in (2.9), are bounded on Lp (RN ) for 1 < p < ∞. It is a general
fact that if Ti is a bounded operator on Lp (Xi , µi ), then T = T1 ⊗ · · · ⊗ Tk is bounded on
Lp (X1 ×, ×Xk , µ1 × · · · × µk ).
56
CHAPTER III
Theorem 2.3. For 1 < p < ∞, kSf kp ≤ C(ψ, p)kf kp . If, in addition, the ψi
satisfy (2.2), f and Sf have equivalent Lp -norms.
Proof. The proof is based on an iteration argument. We only discuss the case when
(2.2) is satisfied.
(j)
For each i and j, let ψ̃i be the measure on RN obtained by tensoring the
(j)
function ψi (xi ) on Rni with the Dirac measure δ0 on the other Rni0 .
(j )
Consider the operator T1 mapping f into the sequence {f ∗ ψ̃1 1 }j1 ∈Z . For a.e.
(x2 , . . . , xk ), denoting by ∗Rm convolution in Rm , we have
(j ) (j ) f ∗RN ψ̃1 1 (x) = f (·, x2 , . . . , xk ) ∗Rn1 ψ1 1 (x1 ) ,
so that, by Theorem 2.2,
Z
R n1
X
p2
Z
2
(j
)
f ∗ ψ̃ 1 (x)
dx1 ∼
1
j1 ∈Z
R n1
f (x)p dx1 .
Integratingin the other variables, we obtain that T1 is bounded from Lp (RN ) to
L RN , `2 (Z) , with equivalence between the norm of f and T1 f .
Consider next T2 , mapping a sequence {gj1 }j1 ∈Z into the double sequence {gj1 ∗
(j2 )
as before shows that T2 is bounded from
ψ̃2 }(j1 ,j2 )∈Z2 . The same argument
p
N 2
p
N 2
2
L R , ` (Z) to L R , ` (Z ) , with equivalence between the norms of {gj1 }
and T2 {gj1 } .
Iterating this argument k times and considering the composition Tk Tk−1 · · · T1 ,
we obtain that
p
Z
RN
X
p
Z
2 2
(j
)
(j
)
1
k
f ∗ ψ̃
dx ∼
∗ · · · ∗ ψ̃k (x)
1
j∈Zk
RN
(j )
f (x)p dx .
(j )
But this is the required estimate, because ψ̃1 1 ∗ · · · ∗ ψ̃k k = ψ (J ) . 3. Marcinkiewicz multipliers
Given s = (s1 , . . . , sk ) ∈ Rk , we define the product Sobolev space H s (RN ) as the
space of tempered distributions f such that fˆ is locally integrable and
kf k2H s
=
Z
RN
1 + kτ1 k2
where, as before, τi ∈ Rni and
have the following analogues.
s1
· · · 1 + kτk k2
i=1
ni = N . Lemmas 6.1 and 6.2 of Chapter II
Pk
sk
|fˆ(τ )|2 dτ < +∞ ,
Lemma 3.1. If si ∈ N for every i, H s (RN ) consists of the L2 -functions f such
that ∂xα11 · · · ∂xαkk f ∈ L2 (RN ) for every choice of the multi-indices αi with |αi | ≤ si
for every i.
Lemma 3.2. Let ϕ be such that ϕ̂ ∈ H s (RN ) with si >
ϕ ∈ L1 (RN ) and, if 0 ≤ ε < si − n2i for every i,
Z
RN
|ϕ(x)| 1 + kx1 k
ε
· · · 1 + kxk k
ε
57
ni
2
for every i. Then
dx ≤ Cε kϕ̂kH s .
In particular, under these hypotheses on s, H s (Rn ) ⊂ C0 (Rn ).
The proofs are direct adaptations of those given in Chapter II.
On each Rni we fix a homogeneous norm (denoted by | | for each i), and a
function ηi ∈ D(Rni ) such that
(i) supp ηi ⊆ {ξi : a0 ≤ |ξi | ≤ b0 },
(ii) ηi (ξi ) ≥ 0 for every ξi e ηi (ξi ) > 0 for a1 ≤ |ξi | ≤ b1 ,
with 0 < a0 < a1 < b1 < b0 are given constants.
We set η(ξ) = η1 (ξ1 ) · · · ηk (ξk ). For r ∈ (R+ )k , we also set mr (ξ) = m(r · ξ),
with the notation of (2.7).
Definition. A Marcinkiewicz multiplier on RN , adapted to the k-parameter dilations (2.7), is a function m such that
sup kmr ηkH s = kmkMs < ∞ ,
(3.1)
r∈(R+ )k
for some s with si >
ni
2
for every i.
We shall denote by Ms (RN ) the class of such multipliers. A simpler pointwise
condition implying that m is a Marcinkiewicz multiplier is that for some s ∈ Nk
with si > n2i for every i, and for every ξ ∈ RN with ξi 6= 0 for every i,
(3.2)
α
∂ 1 · · · ∂ αk m(ξ) ≤ Cα |ξ1 |−d1 (α1 ) |ξ2 |−d2 (α2 ) · · · |ξk |−dk (αk ) ,
ξ1
ξk
where the di are the degrees of the multi-indices αi w.r. to the dilations in Rni .
Condition (3.1) does not depend on the choice of the ηi , and we shall choose the
ηi in such a way that for every i
X
(3.3)
j∈Z
ηi (2j · ξi ) > 0
for ξi 6= 0.
Theorem 3.3. Let m be a Marcinkiewicz multiplier on RN . Then the operator
Sm = m(i−1 ∂) is bounded on Lp (RN ) for 1 < p < ∞.
Qk
Proof. Let ψi = F −1 ηi , ψ(x) = i=1 ψi (xi ). We write 2J ·ξ for (2j1 ·ξ1 , . . . , 2jk ·ξk ),
with J ∈ Zk . Define
mJ (ξ) = m(2−J · ξ)η(ξ) ,
and KJ = F −1 mJ . The Marcinkiewicz condition implies that
(3.4)
Z
RN
|KJ (x)|
2
k
Y
i=1
1 + kxi k2
si
dx ≤ C
58
CHAPTER III
uniformly in J .
Let
(J ) = f ∗ K (J ) ,
TJ f = (Sm f ) ∗ ψ (J ) = Sm (f ∗ ψ (J ) ) = F −1 fˆmψd
J
where, consistently with our previous notation and calling Qi the homogeneous
dimension of Rni w.r. to the given dilations,
(J )
KJ (x) = 2−Q1 j1 · · · 2−Qk jk KJ (2−j1 · x1 , . . . , 2−jk xk ) .
Since each ψi ∗ ψi satisfies (2.1) and (2.2), we have, assuming 2 ≤ p < ∞,
2
X
1
|(Sm f ) ∗ ψ (J ) ∗ ψ (J ) |2 ) 2 kSm f k2p ≤ C p
J ∈Zk
X
1 2 2 2
(J
)
TJ (f ∗ ψ )
= C
p
J ∈Zk
X
TJ (f ∗ ψ (J ) )2 .
= C
p/2
J ∈Zk
Call fJ = f ∗ ψ (J ) and w(x) =
and (3.4),
Qk
i=1
1 + kxi k2
−si
. By Hölder’s inequality
Z
2
2 (J
)
TJ fJ (x) = K
(x
−
y)f
(y)
dy
J
N J
R
Z
Z (J )
2
2
KJ (x − y)
(J )
dy
w (x − y) fJ (y) dy
≤
w (J ) (x − y)
RN
≤ C |fJ |2 ∗ w (J ) (x) .
0
Take g ∈ L(p/2) (RN ). Then
Z
RN
X X Z
TJ fJ (x)2 g(x) dx ≤ C
J ∈Zk
J ∈Zk
=C
X Z
J ∈Zk
≤C
Z
RN
RN
RN
|fJ |2 ∗ w (J ) (x)|g(x)| dx
|fJ (x)|2 w (J ) ∗ |g| (x) dx
X
J ∈Zk
|fJ (x)|
X
2
|f
|
≤ C
J
J ∈Zk
If we prove that the maximal operator
(3.5)
p/2
2
J ∈Zk
(J )
sup w ∗ |g|
J ∈Zk
Mw f (x) = sup w (J ) ∗ |f | (x)
J ∈Zk
sup w (J ) ∗ |g| (x) dx
(p/2)0
.
59
0
is bounded on L(p/2) (RN ), it follows that
X 2 X 2 fJ (x) TJ fJ (x) ≤ C
p/2
J ∈Zk
and we can conclude that
kSm f k2p
J ∈Zk
p/2
X 2 TJ fJ (x) ≤ C
p/2
J ∈Zk
X fJ (x)2 ≤ C 0
p/2
J ∈Zk
X
1 2 2 2
fJ (x)
=C
≤
0
p
J ∈Zk
C 00 kf k2p .
We have used the fact that fj = f ∗ ψ (J ) and that the ψi satisfies (2.1).
The proof of Lp -boundedness of (3.5) is given as a separate lemma. Lemma 3.4. The maximal operator Mw is bounded on Lp (RN ) for 1 < p ≤ ∞.
Proof. We need at this stage to introduce scalar coordinates x = (t1 , . . . , tN ) in RN ,
and we do this in such a way that the first n1 coordinates determine the component
of x in Rn1 , etc. Following the notation used in the rest of this chapter, this means
that
(t1 , . . . , tn1 ) = x1 ,
······
(tN −nk +1 , . . . , tN ) = xk .
Inside each Rni we choose the coordinates so that the dilations are diagonal.
This implies that for each index ` ∈ {1, . . . , N } there are i = i(`) ∈ {1, . . . , k} and
λ` > 0 such that
2J · x = (2ji(`) λ` x` )`=1,...,N .
Write si =
ni
(1
2
+ εi ) with εi > 0. Then, taking i = 1 to simplify the notation,
1 + kx1 k
2 s1
≥
n1
Y
`=1
1 + |t` |
2
21 (1+ε1 )
≥C
n1
Y
`=1
1 + |t` |
1+ε1
.
Therefore, if ε = mini εi ,
w(x) =
k
Y
i=1
1 + kxi k
2 −si
≤C
N
Y
`=1
1 + |t` |
−1−ε
= C w̃(x) .
Therefore Mw f (x) ≤ CMw̃ f (x) for every f and every x, where
Mw̃ f (x) = sup w̃ (J ) ∗ |f | (x)
J ∈Zk
= sup
J ∈Zk
≤ sup
J ∈ZN
Z
N
Y
RN `=1
Z
N
Y
2−ji(`) λ` 1 + |2−ji(`) λ` t0` |
RN `=1
2−j` λ` 1 + |2−j` λ` t0` |
−1−ε f (t1 − t01 , . . . , tN − t0N ) dt0
−1−ε f (t1 − t01 , . . . , tN − t0N ) dt0 .
60
CHAPTER III
Consider first the one-dimensional integral
Z
−1−ε f (t − t0 ) dt0 .
Ij (t) =
2−jλ 1 + |2−jλ t0 |
R
We decompose the real line as the union of the interval [−2jλ , 2jλ ], where 1 +
0
|2−jλ t0 | can be bounded by 1 from below, and the outer dyadic regions {t : 2j λ <
0
0
|t0 | < 2(j +1)λ }, where 1 + |2−jλ t0 | can be bounded from below by 2(j −j)λ . Each
partial integral can be estimated by the Hardy-Littlewood maximal function M f ,
using the trivial inequality
Z
|f (t − t0 )| dt0 ≤ 2rM f (t) .
|t0 |<r
Therefore
Ij (t) ≤ 2
−jλ
+
Z
|t0 |<2jλ
X
2
−jλ
j 0 ≥j
f (t − t0 ) dt0
Z
2j
0λ
<|t0 |<2(j
≤ 2 · 2−jλ 2jλ M f (t) + 2
≤ Cλ,ε M f (t) .
X
0 +1)λ
2(1+ε)(j−j
2−jλ 2(j
0
0
)λ f (t − t0 ) dt0
+1)λ (1+ε)(j−j 0 )λ
2
M f (t)
j 0 ≥j
For each `, define M` f as the one-dimensional Hardy-Littlewood maximal function of f regarded as a function of t` only:
1
M` f (t1 , . . . , tN ) = sup
a<t` <b b − a
Z
b
a
|f (t1 , . . . , t`−1 , t, t`+1 , . . . , tN )| dt .
The estimate obtained for Ij (t) implies that
Z Y
N
R `=1
2−j` λ` 1 + |2−j` λ` t0` |
≤C
N
Y
`=2
−1−ε f (t1 − t01 , . . . , tN − t0N ) dt01 ≤
2−j` λ` 1 + |2−j` λ` t0` |
−1−ε
M1 f (t1 , t2 − t02 . . . , . . . , tN − t0N ) .
Integrating one variable at the time, we find that
Z
N
Y
RN `=1
2−j` λ` 1 + |2−j` λ` t0` |
−1−ε f (t1 − t0 , . . . , tN − t0 ) dt0 ≤
1
N
≤ CMN MN −1 · · · M1 f (x) .
Taking the supremum over J ∈ Zk , we obtain that
Mw f (x) ≤ CMw̃ f (x) ≤ C 0 MN MN −1 · · · M1 f (x) .
61
By Corollary 2.5 in Chapter II, if 1 < p ≤ ∞,
Z +∞
Z +∞
p
|f (x)|p dx` ,
M` f (x) dx` ≤ C
−∞
−∞
for a.e. choice of the x`0 with `0 6= `. Integrating in the remaining variables, we
obtain that
kM` f kp ≤ Ckf kp .
Therefore
kMw f kp ≤ CkMN MN −1 · · · M1 f kp ≤ C 0 kf kp .
4. Applications
We discuss two different applications of the product theory to differential operators. One is to estimates of the kind given in Theorem 7.6 of Chapter II, involving
fractional powers of differential operators, the other is to multipliers of a system of
constant coefficient differential operators.
If L = P (i−1 ∂) is a constant coefficient differential operator on Rn , define, for
γ ∈ C,
|L|γ f = F −1 (|P |γ fˆ) .
In particular, the fractional derivative of order γ in xj (xj being a scalar component of x) is
Djγ f = F −1 (|ξj |γ fˆ) .
The statement of Theorem 7.6 in Chapter II can then be completed as follows.
Theorem 4.1. In the notation of Section 7 in Chapter
II, assume that P satisfies
P
0
(1) and (2 ), and let α1 , . . . , αn ∈ C be such that j λj <eαj ≤ k. Then for every
f ∈ S(Rn ) and 1 < p < ∞,
kD1α1 · · · Dnαn f kp ≤ Cp kf kp + kLf kp .
Proof. We give the proof for n = 2, the general case being essentially the same,
with the extra disadvantage of more complicated notations.
Let ϕ ∈ D(R) be equal to 1 on a neighborhood of 0, and write the Fourier
transform of D α f = D1α1 · · · Dnαn f as
α f (ξ) = |ξ |α1 ϕ(ξ ) |ξ |α2 ϕ(ξ ) fˆ(ξ)
[
D
1
1
2
2
|ξ1 |α1 1 − ϕ(ξ1 )
+
|ξ2 |α2 ϕ(ξ2 ) P (ξ1 , 0)fˆ(ξ)
P (ξ1 , 0)
|ξ2 |α2 1 − ϕ(ξ2 )
α1
P (0, ξ2 )fˆ(ξ)
+ |ξ1 | ϕ(ξ1 )
P (0, ξ2 )
|ξ1 |α1 1 − ϕ(ξ1 ) |ξ2 |α2 1 − ϕ(ξ2 )
ˆ
+
P (ξ)f(ξ)
P (ξ)
ˆ + m2 (ξ)P (ξ1, 0)fˆ(ξ) + m3 (ξ)P (0, ξ2)fˆ(ξ) + m4 (ξ)P (ξ)f(ξ)
ˆ .
= m1 (ξ)f(ξ)
62
CHAPTER III
We begin with m2 and verify condition (3.2), taking s1 = s2 = 1. Because m2 is
the product of two factors in splitted variables, we need to observe that
(1) m2 is bounded (the hypotheses on P imply that |P (ξ1 , 0)| ≥ C|ξ1 |k/λ1 );
(2) the derivative of |ξ2 |α2 ϕ(ξ2 ) is bounded by a constant times |ξ2 |−1 for ξ2 6= 0;
|ξ1 |α1 1−ϕ(ξ1 )
(3) the derivative of
is smooth and, away from the support of ϕ,
P (ξ1 ,0)
it is homogeneous (in the ordinary sense) of degree one less than the degree
of this fraction itself (this is α1 − λk1 , which has a non-positive real part);
hence the derivative is bounded by a constant times |ξ1 |−1 for ξ1 6= 0.
The same remarks imply that also m1 and m3 satisfy (3.2) with s1 = s2 = 1.
The verification that m4 also verifies (3.2) with s1 = s2 = 1 cannot be done by
separation of variables.
Observe that m4 equals 1 − ϕ(ξ1 ) 1 − ϕ(ξ2 ) times a function which is continuous and homogeneous, w.r. to the non-isotropic dilations, of degree λ1 α1 +λ2 α2 −k,
whose real part is non-positive. Since ξ1 and ξ2 are bounded away from 0 on the
support of m4 , we conclude that m4 is bounded.
We skip now to estimating that ∂ξ1 ∂ξ2 m4 is bounded by |ξ1 |−1 |ξ2 |−1 times a
constant. We apply Leibniz’s rule.
If both derivatives fall on 1 − ϕ(ξ1 ) 1 − ϕ(ξ2 ) , this term is supported on
a compact set and the estimate is trivial. Consider then the case where the ξ1 derivative falls on 1 − ϕ(ξ1 ) and the ξ2 -derivative on the homogeneous function.
The derivation in ξ2 reduces the non-isotropic degree of homogeneity by λ2 . Hence
we obtain −ϕ0 (ξ1 ) times a function whose non-isotropic degree of homogeneity is
λ1 α1 + λ2 (α2 − 1) − k, whose real part is ≤ −λ2 .
Considering that this term is supported on a set where a < |ξ1 | c
for appropriate a, b, c > 0, this term is bounded by a constant times
−λ2
∼ |ξ2 |−1 ∼ |ξ1 |−1 |ξ2 |−1 .
|ξ|−λ2 ∼ |ξ1 |1/λ1 + |ξ2 |1/λ2
The same applies if the rôle of ξ1 and ξ2 is interchanged. Suppose finally that
both derivatives fall on the homogeneous factor. We then obtain a function homogeneous of degree λ1 (α1 − 1) + λ2 (α2 − 1) − k, whose real part is ≤ −λ1 − λ2 .
Considering that the other factor restrict the support to |ξ1 |, |ξ2 | > c > 0, we get
the bound
−λ1 −λ2
|ξ|−λ1 −λ2 ∼ |ξ1 |1/λ1 + |ξ2 |1/λ2
≤ C|ξ1 |−1 |ξ2 |−1 .
The remaining verifications follow the same lines. Therefore all the mi are
Marcinkiewicz multipliers. If we call L1 = P (i−1 ∂x1 , 0), and L2 P (0, i−1 ∂x2 ), we
have proved that
kD α f kp ≤ Cp kf kp + kL1 f kp + kL2 f kp + kLf kp ,
for 1 < p < ∞. But we can nowapply Theorem 7.6 in Chapter II directly to show
that kLi f kp ≤ C kf kp + kLf kp for i = 1, 2. We pass now to the second application. On Rn are given k differential operators
Li = Pi (i−1 ∂), where
(1) for each i there is a subspace Vi of Rn of dimension ni ≤ n such that
Pi (ξ) = Pi (ξi ), where ξi is the orthogonal projection of ξ on Vi ;
63
(2) Pi (ξi ) > 0 for ξi ∈ Vi \ {0};
(3) each Pi is homogeneous of degree ki w.r. to some non-isotropic 1-parameter
dilations on Vi .
Theorem 4.2. Let m(λ1 , . . . , λk ) be a Marcinkiewicz multiplier on Rk = R × · · · ×
R, with the standard dilations on each coordinate line, of order s = (s 1 , . . . , sk ) with
si > n2i for every i. Then m(L1 , . . . , Lk ) is bounded on Lp (Rn ) for 1 < p < ∞.
We point out that this is not the optimal result, but is what can be obtained
by our method of proof. The process of lifting to a higher–dimensional space, as
described below, forces us to impose the Marcinkiewicz conditions adapted to the
higher dimension.
Proof. On each Vi we fix coordinates that reduce the dilations to diagonal form.
For each i, this choice of coordinates determines a linear bijection τi : Rni → Vi .
Let P̃i = Pi ◦ τi .
On Rn1 × · · · × Rnk = RN we consider the operators
L̃i = P̃i (i−1 ∂xi ) ,
where ∂xi is the gradient in Rni ∼ Vi . For convenience, we use the symbol ηi ∈ Rni
to denote the variable for Fourier transforms on Rni .
We claim that the Fourier multiplier on RN
m̃(η1 , . . . , ηk ) = m P̃1 (η1 ), . . . , P̃k (ηk )
is a Marcinkiewicz multiplier of order s w.r. to the k-parameter dilations on RN
induced by the 1-parameter dilations on each Rni (we skip the details of the proof,
which is a refinement of the proofs of Theorem 7.1 and Proposition 7.4 in Chapter
II). Therefore m(L̃1 , . . . , L̃k ) is bounded on Lp (RN ) for 1 < p < ∞.
Call σi = τi−1 ◦ πi : Rn → Rni , where πi is the orthogonal projection onto Vi in
Rn , and σ : Rn → RN the map
σ(ξ) = σi (ξ), . . . , σk (ξ) .
Then the Fourier multiplier of m(L1 , . . . , Lk ) is
m P1 (ξ), . . . , Pk (ξ) = m̃ ◦ σ(ξ) .
We give the conclusion after the next lemma. What follows is part of a general principle, called transference, whose general
idea is that an Lp -estimate for a convolution operator (or, equivalently, for a Fourier
multiplier) induces other Lp -estimates for a certain class of induced (or transferred)
operators. The statement we give concerns transference from Euclidean space to
another, by means of a linear map.
Lemma 4.3. Let σ : Rn → Rd be a linear map, and let m be a Fourier Lp multiplier on Rd , continuous on the image of A. Then m ◦ σ is a Fourier Lp multiplier on Rn , and, if Sm and Sm◦σ are the corresponding operators,
kSm◦σ kL(Lp (Rn )) ≤ kSm kL(Lp (Rd )) .
Proof. Changing coordinates on Rd and Rn if necessary, we can assume that
0
(1) Rd = Rν × Rd , and im σ = Rν × {0};
0
0
(2) Rn = Rν × Rn , and ker σ = {0} × Rn .
64
CHAPTER III
Introducing coordinates (η, η 0 ) on Rd and (ξ, ξ 0) on Rn compatible with these
splittings, we then have σ(ξ, ξ 0) = (Aξ, 0) for some invertible ν × ν matrix A.
The proof follows from the combination of three facts.
First fact: if m(η, η 0 ) is a Fourier Lp -multiplier on Rd , continuous on Rν × {0},
then m0 (η, η 0 ) = m(η, 0) is also a Fourier Lp -multiplier on Rd , with no increase in
the norm. In order to see this, for ε > 0 consider mε (η, η 0 ) = m(η, εη 0 ). Then, for
f, g ∈ S(Rd ),
Z
d
2
m(η, εη 0 )fˆ(η, η 0 )ĝ(η, η 0 ) dη dη 0
hSmε f |gi = (2π)
d
ZR
0
d0
d
−d
m(η, η 0 )ε p0 fˆ(η, ε−1 η 0 )ε− p ĝ(η, ε−1 η 0 ) dη dη 0
= (2π) 2
Rd
= hSm fε |gε i ,
d0
d0
where fε (y, y 0 ) = ε p f (x, εx0 ) and gε (y, y 0) = ε p0 g(x, εx0 ). Therefore
hSmε f |gi ≤ kSm kL(Lp (Rd )) kfε kp kgε kp0 = kSm kL(Lp (Rd )) kf kp kgkp0 .
If ε tends to 0, hSmε f |gi tends to hSm0 f |gi, and this proves the first fact.
Second fact: a bounded function m(η, η 0 ) = µ(η) is a Fourier Lp -multiplier on Rd
if and only if µ is a Fourier Lp -multiplier on Rν and the two norms coincide. To see
this, let k = F −1 µ ∈ S 0 (Rν ). Since m = µ ⊗ 1, we have F −1 m = k ⊗ δ0 ∈ S 0 (Rd ).
0
Calling f y (y) = f (y, y 0 ), we then have
0
0
Sm f (y, y 0) = f ∗Rd (k ⊗ δ0 )(y, y 0 ) = f y ∗Rν k(y) = Sµ f y (y) .
Therefore,
kSm f kpLp (Rd )
Z
0
kSµ f y kpLp (Rν ) dy 0
0
d
R
Z
0
p
≤ kSµ kL(Lp (Rν ))
kf y kpLp (Rν ) dy 0
=
0
≤
Rd
p
kSµ kL(Lp (Rν )) kf kpp
.
This proves that kSm f kLp (Rd ) ≤ kSµ kL(Lp (Rν )) . For the opposite inequality, take
f (y, y 0) = f1 (y)f2 (y 0 ). Then Sm f (y, y 0 ) = (Sµ f1 )(y)f2 (y 0 ). With f2 6= 0 fixed,
kSµ f1 kp =
kSm f kp
≤ kSm f kLp (Rd ) kf1 kp .
kf2 kp
Third fact: the statement is true if d = n = ν. In This case m̃(ξ) = m(Aξ) with
A invertible. Taking again f, g ∈ S(Rν ), we have
Z
ν
ˆ
2
hSm̃ f |gi = (2π)
m(Aξ)f(ξ)ĝ(ξ)
dξ
Rν
Z
ν
−1
2
m(η)fˆ(A−1 η)ĝ(A−1 η) dη
= (2π) | det A|
Rν
= | det A|
−1
hSm fA |gA i ,
65
where fA (x) = | det A|f (Ax), gA (x) = | det A|g(Ax). Then
hSm̃ f |gi ≤ | det A|−1 kSm kL(Lp (Rν )) kfA kp kgA kp0 = kSm kL(Lp (Rν )) kf kp kgkp0 .
Going back to the general statement, from the Lp -boundedness of Sm on Rd
we deduce the Lp -boundedness of µ(η) = m(η, 0) on Rν (facts 1 and 2). From
this we deduce the Lp -boundedness of Sµ◦A on Rν (fact 3), and from this the
Lp -boundedness of Sm◦σ on Rn (fact 2 again). End of the proof of Theorem 4.2. Lemma 4.3 implies that m(L1 , . . . , Lk ) is bounded
on Lp for 1 < p < ∞ when m is a Marcinkiewicz multiplier, continuous on Rk . Let
m be a generic Marcinkiewicz multiplier on Rk , and observe that Lemma 3.2 implies
that m is continuous when λi 6= 0 for every i.
Given ε > 0 and ϕ ∈ D(R) with supp ϕ ⊂ [−2, 2] and ϕ = 1 on [−1, 1], let
mε (λ) = m(λ)
k Y
i=1
λi 1−ϕ
.
ε
Then mε is continuous. If we prove that the Marcinkiewicz norm (3.1) of
the mε is uniformly bounded in ε, we can conclude that the operator norms
of mε (L1 , . . . , Lk ) are uniformly bounded. It is then easy to observe, using the
Plancherel formula, that, for f, g ∈ S(Rn ),
hm(L1 , . . . , Lk )f |gi = lim hmε (L1 , . . . , Lk )f |gi ,
ε→0
which would allow to conclude. Fix η ∈ D(R) satisfying (i) and (ii) in Section 3.
We must estimate the H s -norm of
m(r1 λ1 , . . . , rk λk )
k Y
i=1
r i λi Y
η(λi ) .
1−ϕ
ε
i=1
k
Observe that 1 − ϕ(ri λi /ε) η(λi ) is identically zero if ri /ε ≤ 1/4, and it equals
η(λi ) if ri /ε ≥ 4. Matters reduce therefore to verifying that multiplication by
ϕ(tλi ) is a continuous operation on H s , uniformly for 1/4 ≤ t ≤ 4. This fact can
be proved by adapting the proof of Lemma 6.3 in Chapter II. 66
CHAPTER III
HEISENBERG GROUP
67
CHAPTER IV
FOURIER ANALYSIS ON THE HEISENBERG GROUP
1. The Heisenberg group
On R2n+1 = Rn × Rn × R consider the composition law
1
(x, y, t)(x0, y 0 , t0 ) = x + x0 , y + y 0 , t + t0 + (x · y 0 − y · x0 ) .
2
In complex coordinates (z, t) ∈ Cn × R, this becomes
1
(z, t)(z 0 , t0 ) = z + z 0 , t + t0 − =mhz|z 0 i ,
2
Pn
where hz|z 0 i = j=1 zj zj0 is the Hermitean product on Cn .
One can verify that this is a non-commutative group law, with neutral element
(0, 0), and with inverse (z, t)−1 = (−z, −t). This is the Heisenberg group Hn .
The elements (0, t) form the center Z(Hn ) of Hn , and Hn /Z(Hn ) is isomorphic
to the additive group R2n . The following transformations are automorphisms of
Hn :
(1) the dilations (z, t) 7−→ (rz, r 2 t) = r · (z, t), with r > 0;
(2) the rotations (z, t) 7−→ (U z, t), with U a unitary transformation of Cn (i.e.
U U ∗ = I);
(3) the conjugation (z, t) 7−→ (z̄, −t).
One finds the vector fields Xj , Yj in (2.1), Chapter I, by means of the following
operations:
d
f (x, y, t)(sej , 0, 0)
ds |s=0
1
d
f x + sej , y, t − syj
=
ds |s=0
2
yj
= ∂xj f (x, y, t) − ∂t f (x, y, t) ,
2
d
Yj f (x, y, t) =
f (x, y, t)(0, sej , 0)
ds |s=0
1
d
f x, y + sej , t + sxj
=
ds |s=0
2
xj
= ∂yj f (x, y, t) + ∂t f (x, y, t) .
2
Xj f (x, y, t) =
(1.1)
Typeset by AMS-TEX
68
CHAPTER IV
Therefore these vector fields are left-invariant: denoting by
L(z 0 ,t0 ) f (z, t) = f (z 0 , t0 )−1 (z, t)
the left-translate of f by (z 0 , t0 ), then
Xj L(z 0 ,t0 ) f = L(z 0 ,t0 ) (Xj f ) ,
It follows that also
Yj L(z 0 ,t0 ) f = L(z 0 ,t0 ) (Yj f ) .
[Xj , Yj ] = T = ∂t
is left-invariant and that any left-invariant vector field on Hn is a linear combination
of the Xj , Yj , T . The other Lie brackets are
[Xj , Xk ] = [Yj , Yk ] = [Xj , T ] = [Yj , T ] = 0 ,
[Xj , Yk ] = δj,k T .
More generally, one says that a linear operator T acting on functions on Hn is
left-invariant if
T (L(z,t) f ) = L(z,t) (T f )
for all (z, t) ∈ Hn and every f . We impose here and in the sequel that T maps13
S(Hn ) into S 0 (Hn ) boundedly, in the sense that the map14
(f, g) 7−→ hT f, gi
is continuous on S(Hn ) × S(Hn ). This is a very mild initial assumption. It is
satisfied, e.g., by the convolution operators
(1.2)
T f = f ∗ k(z, t) =
Z
Hn
f (z, t)(w, u)−1 k(w, u) dw du ,
with k ∈ L1 (Hn ) (which must be understood w.r. to the Lebesgue measure).
Convolution on Hn is a non-commutative operation, and in order to have a leftinvariant operator, the kernel must be on the right of f . The convolution (1.1) can
be extended to distributional kernels k ∈ S 0 (Hn ), provided f ∈ S(Hn ); the integral
on the right-hand side of (1.2) must then be
interpreted as the pairing between k
−1
and the function (w, u) 7−→ f (z, t)(w, u)
.
It follows from the Schwartz kernel theorem that any left-invariant operator T
mapping S(Hn ) boundedly into S 0 (Hn ) has the form (1.2) for some k ∈ S 0 (Hn ).
Pn
The sub-Laplacian L = − j=1 (Xj2 + Yj2 ) studied in Chapter I is also leftinvariant, and such are its spectral projections, the joint spectral projections of
L and i−1 T , and the operators defined by spectral multipliers of L and i−1 T .
Therefore they can all be realized as convolution operators on Hn .
13 S(H
2n+1 ), and similarly for S 0 .
n ) is the same as S(R
14 Recall that the notation h , i stands for the bilinear pairing.
HEISENBERG GROUP
69
Obviously, besides the left-invariant vector fields, one has the right-invariant
ones. We shall put a superscript (r) to denote the right-invariant versions of
the Xj , Yj :
d
f (sej , 0, 0)(x, y, t)
ds |s=0
1
d
f x + sej , y, t + syj
=
ds |s=0
2
yj
= ∂xj f (x, y, t) + ∂t f (x, y, t) ,
2
d
(r)
Yj f (x, y, t) =
f (0, sej , 0)(x, y, t)
ds |s=0
1
d
f x, y + sej , t − sxj
=
ds |s=0
2
xj
= ∂yj f (x, y, t) − ∂t f (x, y, t) .
2
(r)
Xj f (x, y, t) =
(1.3)
Observe that T is both right- and left-invariant, and that
(r)
(r)
[Xj , Yj ] = −T .
We shall also write
L(r) = −
n
X
j=1
(r)
(r) (Xj )2 + (Yj )2 .
One easily verifies that, if fˇ(z, t) = f (z, t)−1 = f (−z, −t), then
(r)
Xj f = −(Xj fˇ)ˇ,
(similarly for the Yj ) and that
L(r) f = (Lfˇ)ˇ.
2. The group Fourier transform
For a few sections we shall restrict ourselves to H1 = C × R. We recall that in
Chapter I we have introduced the functions
(2.1)
hj,k (z) = Z̄ k (z̄ j e−
|z|2
4
)=e
|z|2
4
∂z̄k (z̄ j e−
|z|2
2
)
on C, with j, k ∈ N, and discussed their rôle in the spectral analysis of the subLaplacian L. As we shall see, these functions are the key ingredients to construct
the group Fourier transform on H1 (later on we shall extend all this to Hn ).
The general notion of Fourier transform on locally compact groups involves notions of abstract representation theory that we do not want to develop here 15 .
15 See
the notes of the course “Analisi di Fourier non commutativa”.
70
CHAPTER IV
We shall use instead the spectral analysis of L to study the specific case of the
Heisenberg group.
To begin with, it is appropriate to normalize the hj,k in L2 (C). From (3.8) in
Chapter I, we deduce that
k!
khj,0 k2
2k Z 2
∞
r2
k!
= k 2π
r 2j+1 e− 2 dr
2
0
khj,k k22 =
= (2π)j!k!2j−k .
We then define
1
ϕj,k (z) = p
hj,k (z) .
j!k!2j−k
(2.2)
By Proposition 3.2 in Chapter 1, {(2π)−1/2 ϕj,k } is an orthonormal basis of
L2 (C). From (3.10) in Chapter I, it follows that the ϕj,k are in S(C).
By (3.10) and (3.12) in Chapter I,
(2.3)
ϕj,k (0) = δj,k ,
ϕj,k (z) = (−1)j−k ϕj,k (−z) = (−1)j−k ϕk,j (z̄) = (−1)j−k ϕk,j (z) .
In analogy with Corollary 3.3 in Chapter I, we define

1
if λ > 0 ,
 ϕj,k (λ 2 z)
λ
(2.4)
ϕj,k (z) =
1 
if λ < 0 .
ϕj,k |λ| 2 z̄
We also set
Φλj,k (z, t) = eiλt ϕλj,k (z) .
(2.5)
Given f ∈ L1 (H1 ), we define16 , for λ 6= 0,
Z
ˆ
(2.6)
f (λ, j, k) =
f (z, t)Φλj,k (z, t) dz dt = hFt f (·, −λ)|ϕλj,k i .
H1
Proposition 2.1. If f ∈ L1 (H1 ) ∩ L2 (H1 ), then the following Plancherel formula
holds:
Z +∞ X
1
2
fˆ(λ, j, k)2 |λ| dλ .
(2.7)
kf k2 =
2
(2π) −∞
j,k∈N
The map f 7−→ (2π)−1 fˆ extends to an isometric bijection from L2 (H1 ) onto
L R, |λ| dλ, `2(N2 ) .
2
16 It
will be clear soon (Theorem 2.3) that it is preferable not to take complex conjugates of
the basis elements.
HEISENBERG GROUP
71
Proof. By the Plancherel formula on R,
Z
1
|f (z, t)| dt =
2π
H1
2
Z
Z
+∞
−∞
C
Ft f (z, −λ)2 dz dλ .
1
Then, for a.e. λ, Ft f (z, −λ) ∈ L2 (C). But (2π)−1/2 |λ| 2 ϕλj,k j,k is an orthonormal basis of L2 (C); hence
Z
X Ft f (z, −λ)2 dz = |λ|
hFt f (·, −λ) | ϕλ i2
j,k
2π
C
j,k∈N
=
2
|λ| X ˆ
f (λ, j, k) .
2π
j,k∈N
We shall give a more elegant presentation of this result, introducing at the same
time the representations of H1 .
Using (2.1) above and the identity (3.6) in Chapter I, we obtain the following
identities:
r
k+1
ϕj,k+1 ,
Z̄ϕj,k =
2
(2.8)
r
r
r
2
k
1 2
Zϕj,k =
Z Z̄ϕj,k−1 = −
(L + I)ϕj,k−1 = −
ϕj,k−1 .
k
4 k
2
Lemma 2.2. The following identity holds:
X
i
ϕj,` (z)ϕ`,k (w) = e− 2 =m(zw̄) ϕj,k (z + w) .
`∈N
Proof. By (3.9) in Chapter I and by (2.3) above,
X
ϕj,` (z)ϕ`,k (w) =
`∈N
X
(−1)j−` ϕ`,j (z)ϕ`,k (w)
`∈N
= (−1)
= (−1)
The series
sum equals
P
j
s
2j+k X (−1)`
h`,j (z)h`,k (w)
j!k!
`!2`
`∈N
2j+k |z|2 +|w|2 X (−1)` j ` − |z|2 k ` − |w|2 e 4
∂ z e 2 ∂w̄ w̄ e 2 .
j!k!
`!2` z
(−1)` ` − |z|
2
`∈N `!2` z e
X (−1)`
`∈N
j
s
`!2`
z ` e−
|z|2
2
`∈N
2
w̄ ` e−
w̄ ` e−
|w|2
2
|w|2
2
converges with all its derivatives. The
= e−
|z|2 +|w|2 +z w̄
2
= e−
|z+w|2 −z̄w
2
.
72
CHAPTER IV
Therefore,
X
ϕj,` (z)ϕ`,k (w) = (−1)j
`∈N
s
2j+k |z|2 +|w|2 j k − |z+w|2 −z̄w
2
∂z ∂w̄ e
e 4
j!k!
|z+w|2 |z|2 +|w|2 +2z̄w
1
k
4
(z + w)j e− 2
∂w̄
= p
e
j!k!2j−k
|z+w|2 |z+w|2 +2i=mz̄w
1
k
4
= p
e
∂w̄
(z + w)j e− 2
j!k!2j−k
1
1
= p
e− 2 =mzw̄ hj,k (z + w)
j!k!2j−k
1
= e− 2 =mzw̄ ϕj,k (z + w) .
It follows immediately that
(2.9)
X λ
Φj,` (z, t)Φλ`,k (w, u) ,
Φλj,k (z, t)(w, u) =
`∈N
for every λ 6= 0.
It is then natural to arrange the Φλj,k in the infinite matrix
Φλ (z, t) = Φλj,k (z, t)
(2.10)
Theorem 2.3. The following identities hold:
j,k
.
Φλ (z, t)∗ Φλ (z, t) = Φλ (z, t)Φλ (z, t)∗ = I
Φλ (z, t)−1 = Φλ (z, t)∗ = Φλ (z, t)−1 ,
Φλ (z, t)(w, u) = Φλ (z, t)Φλ (w, u) .
In particular, Φλ (z, t) defines, for every (z, t) ∈ H1 and every λ 6= 0, a unitary
operator π λ (z, t) on `2 = `2 (N). The map
π λ : H1 −→ L(`2 )
is a continuous homomorphism, w.r. to the strong topology on L(` 2 ), i.e. a unitary
representation of H1 .
For f ∈ L1 (H1 ), the integral
(2.11)
λ
π (f ) =
Z
f (z, t)π λ (z, t) dz dt
H1
converges in the strong topology to a bounded operator on ` 2 . The identity
(2.12)
π λ (f ∗ g) = π λ (f )π λ (g)
holds for every f, g ∈ L1 (H1 ).
HEISENBERG GROUP
73
The proof requires a few verifications that we leave to the reader. Observe that
the operator π λ (f ) in (2.11) is represented, w.r. to the canonical basis of `2 , by the
matrix
Z
def
f (z, t)Φλ (z, t) dz dt = fˆ(λ, j, k) j,k = fˆ(λ) .
H1
Then (2.12) takes the form
(2.13)
f[
∗ g(λ) = fˆ(λ)ĝ(λ) ,
in analogy with the ordinary Fourier transform. These identities justify the choice of
not conjugating Φλj,k in (2.6). If we had done so, the order of the two factors in (2.13)
should have been changed. Recall that both convolution in H1 and composition of
linear operators on `2 are non-commutative.
We obtain the following reformulation of Proposition 2.1.
Theorem 2.4. For f ∈ L1 (H1 ) ∩ L2 (H1 ), the operator π λ (f ) is a Hilbert-Schmidt
operator for a.e. λ, and the Plancherel formula can be written as:
Z +∞
1
2
(2.14)
kf k2 =
kπ λ (f )k2HS |λ| dλ .
2
(2π) −∞
−1 λ
The map f 7−→ (2π)
π (f ) extends to an isometric bijection from L2 (H1 ) onto
2
L R, |λ| dλ, HS(` ) .
2
This is the standard form of a Plancherel formula on a non-commutative group,
invoking Hilbert-Schmidt norms of operator-valued Fourier transforms.
We write explicitely the polarized form of Plancherel’s formula:
Z
Z +∞
1
f (z, t)g(z, t) dz dt =
tr π λ (f )π λ (g)∗ |λ| dλ
2
(2π) −∞
H1
Z +∞
1
ˆ(λ)ĝ(λ)∗ |λ| dλ
tr
f
=
(2.16)
(2π)2 −∞
Z +∞ X
1
fˆ(λ, j, k)ĝ(λ, j, k) |λ| dλ .
=
(2π)2 −∞
j,k∈N
Like in commutative Fourier analysis, the Plancherel formula has a companion
inversion formula. We give the inversion formula for a narrow class of functions
(the Schwartz class). It can however be extended to a larger class17 . The proof
requires a few lemmas.
Lemma 2.5. For every j, k, kΦλj,k k∞ ≤ 1. If L = −X 2 − Y 2 is the sub-Laplacian
on H1 , and T = ∂t ,
(2.17)
LΦλj,k = |λ|(2k + 1)Φλj,k ,
T Φλj,k = iλΦλj,k .
Proof. The first identity in Theorem 2.3 implies that, for each j and each (z, t),
X
(2.18)
|Φλj,k (z, t)|2 = 1 .
k∈N
This obviously proves the first statement. The first identity in (2.17) is a consequence of Corollary 3.3 in Chapter I, and the second is obvious. λ
formula below
` λ shows
´ that it is sufficient that π (f ) be of trace class for a.e. λ and that
the function λ 7→ tr |π (f )| be integrable on R. The proof below shows that Schwartz functions
satisfy this property.
17 The
74
CHAPTER IV
Lemma 2.6. The following identities hold:
c (λ, j, k) = |λ|(2k + 1)fˆ(λ, j, k) ,
Lf
If f ∗ (z, t) = f (−z, −t) then
Tcf (λ, j, k) = −iλfˆ(λ, j, k) .
fc∗ (λ, j, k) = fˆ(λ, k, j) ,
i.e.
πλ (f ∗ ) = πλ (f )
∗
.
Proof. The first two identities follow directly from Lemma 2.5, by integration by
parts. The last one follows from the fact that Φλj,k (−z, −t) = Φλk,j (z, t). Proposition 2.7. For each N ∈ N, there is a Schwartz norm k kN such that, if
f ∈ S(H1 ),
kf kN
fˆ(λ, j, k) ≤
N .
1 + |λ|(1 + j + k)
Moreover fˆ(λ, j, k) is smooth in λ for λ 6= 0.
Proof. If f ∈ S(H1 ), we have in the first place
Z
λ
fˆ(λ, j, k) = f (z, t)Φj,k (z, t) dz dt ≤ kf k1 kΦλj,k k∞ ≤ kf k1 .
H1
By Lemma 2.6,
N f (λ, j, k) ≤ kT N f k ,
|λ|N fˆ(λ, j, k) = T[
1
[
N f (λ, j, k) ≤ kLN f k ,
|λ|N (2k + 1)N fˆ(λ, j, k) = L
1
|λ|N (2j + 1)N fˆ(λ, j, k) = |λ|N (2j + 1)N c
f ∗ (λ, k, j) ≤ kLN f ∗ k1 ,
Putting all these estimates together,
1 + |λ|(1 + j + k)
N
|fˆ(λ, j, k)| ≤ C kf k1 + kT f k1 + kLf k1 + kL∗ f k1 ,
and this last expression is controlled by a Schwartz norm.
The smoothness of fˆ(λ, j, k) for λ 6= 0 is obvious. Theorem 2.8. For f ∈ S(H1 ) the following inversion formula holds:
(2.19)
1
f (z, t) =
(2π)2
=
1
(2π)2
Z
Z
+∞
−∞
+∞
tr π λ (f )π λ (z, t)∗ |λ| dλ
X
−∞ j,k∈N
fˆ(λ, j, k)Φλj,k (z, t) |λ| dλ .
Proof. Since
fˆ(λ, j, k) = hFt f (·, −λ) | ϕλj,k i ,
HEISENBERG GROUP
75
for every λ 6= 0, and Ft f (·, −λ) ∈ L2 (C),
Ft f (z, −λ) =
|λ| X ˆ
f (λ, j, k)ϕλj,k (z) ,
2π
j,k∈N
with convergence in L2 (C). The convergence is also pointwise. In fact, by (2.18),
|ϕj,k (z)| ≤ 1, so that, by Proposition 2.7,
X
j,k∈N
|fˆ(λ, j, k)||ϕλj,k (z)| ≤
taking N > 2.
Therefore,
Ft f (z, −λ)e−iλt =
X
j,k∈N
kf kN
1 + |λ|(1 + j + k)
N < ∞ ,
|λ| X ˆ
f (λ, j, k)Φλj,k (z, t) ,
2π
j,k∈N
and the Fourier inversion formula on R does the rest.
We give a first application of this formula, which will be used in the sequel.
Corollary 2.9. Let S0 (H1 ) ⊂ S(H1 ) be the space of the functions f such that
fˆ(λ, j, k) is non-zero only for j, k varying in a finite set, and, for these values of
j, k, fˆ(λ, j, k) is C ∞ in λ and supported on a compact subset of R \ {0}. Then
S0 (H1 ) is dense in L2 (H1 ).
Given any finite family of functions vj,k ∈ D(R \ {0}), with (j, k) ∈ B, the
function
Z X
1
vj,k (λ)Φλj,k (z, t) |λ| dλ
f (z, t) =
(2π)2 R
(j,k)∈B
is in S0 (H1 ) and fˆ(λ, j, k) equals vj,k (λ) if (j, k) ∈ B and zero otherwise.
Proof. It is sufficient to prove that any g ∈ S(H1 ) can be approximated in the
L2 -norm by functions in S0 (H1 ). Given ε > 0, take K ⊂ R \ {0} and N ∈ N such
that
Z
X
|ĝ(λ, j, k)|2 |λ| dλ < ε2 ,
R\K j+k>N
and fix η ∈ D(R \ {0}) equal to 1 on K, with 0 ≤ η(λ) ≤ 1 for every λ. Let
K 0 = supp η and define
Z
X
1
ĝ(λ, j, k)Φλj,k (z, t) η(λ)|λ| dλ
fε (z, t) =
2
(2π) K 0
j+k≤N
Z
X
1
=
ĝ(λ, j, k)ϕλj,k (z)η(λ)|λ|e−iλt dλ
(2π)2
0
K
j+k≤N
1 X
Ft−1 ĝ(λ, j, k)ϕλj,k (z)η(λ)|λ| (−t) .
=
2π
j+k≤N
It follows from Proposition 2.7 that ĝ(λ, j, k)ϕλj,k (z)η(λ)|λ| ∈ S(C×R). Therefore
fε ∈ S(H1 ).
76
CHAPTER IV
Moreover,
fε ∈ S0 (H1 ).
The proof
fbε (λ, j, k) = η(λ)ĝ(λ, j, k) if j + k ≤ N , and 0 otherwise. Hence
Finally, by the Plancherel formula, kg − fε k2 < Cε.
of the last statement is now obvious. We conclude this section by discussing the possibility of defining the Fourier
transform for a general distribution u ∈ S 0 (H1 ). In analogy with (2.6), we are
tempted to define û(λ, j, k) = hu, Φλj,k i. But Φλj,k is not in S(H1 ) because it does
not have any decay in t, even though it satisfies the required decay conditions in z.
We can however apply u to a “packet” of Φλj,k .
Given ψ ∈ D(R \ {0}), define
Z
ψ̃j,k (z, t) =
Φλj,k (z, t)ψ(λ) dλ = 2πFt−1 ϕλj,k (z)ψ(λ) (t) .
R
Since
∈ S(C × R) as a function of z and λ, it follows that ψ̃j,k ∈
S(H1 ). We can then defined distributions uj,k ∈ D 0 (R \ {0}) by setting
ϕλj,k (z)ψ(λ)
hûj,k , ψi = hu, ψ̃j,k i .
(2.20)
It is a simple verification that, if u ∈ L1 (H1 ), then ûj,k coincides with fˆ(λ, j, k)
as a function of λ.
The convolution formula in (2.13) has the following extension.
Lemma 2.10. If u ∈ S 0 (H1 ), and f ∈ S0 (H1 ),
X
\
fˆ(·, j, `)û`,k .
(f
∗ u)j,k =
`∈N
Proof. By (2.20), if ψ ∈ D(R \ {0}) and fˇ(z, t) = f (−z, −t),
\
h(f
∗ u)j,k , ψi = hf ∗ u, ψ̃j,k i
= hu, fˇ ∗ ψ̃j,k i .
By (2.9),
Z
f (w, u)ψ̃j,k (w, u)(z, t) dw du
H
Z 1Z
=
f (w, u)Φλj,k (w, u)(z, t) ψ(λ) dλ dw du
H1 R
XZ
=
fˆ(λ, j, `)Φλ`,k (z, t)ψ(λ) dλ
fˇ ∗ ψ̃j,k (z, t) =
`∈N
=
X
`∈N
R
ψ fˆ(·, j, `) ˜`,k .
But, according to (2.20),
ˆ j, `)i = hfˆ(·, j, `)û`,k , ψi ,
u, ψ fˆ(·, j, `) ˜`,k = hû`,k , ψ f(·,
and this concludes the proof.
A complete description of the image of S 0 (H1 ) under Fourier transform is possible18 , but we do not go into it because it will not be needed.
18 See
D. Geller, ....
HEISENBERG GROUP
77
3. Fourier multipliers
c (λ)
It follows from Lemma 2.6 that the matrices fˆ(λ) = fˆ(λ, j, k) j,k and Lf
are related by the identity


|λ|
0
···
0
···
 0 3|λ| · · ·
0
··· 


 ..

..
..
c
ˆ

 .
.
(3.1)
Lf (λ) = f (λ)  .
.

 0

0
(2k
+
1)|λ|


..
..
..
.
.
.
Similarly,
Tcf (λ) = −iλfˆ(λ) = fˆ(λ)(−iλI) .
(3.2)
Similar formulas can be obtained for the other left-invariant vector fields. In
analogy with (3.2) in Chapter I, define the complex vector fields
Z=
i
1
(X − iY ) = ∂z − z̄∂t ,
2
4
Z̄ =
1
i
(X + iY ) = ∂z̄ + z∂t ,
2
4
on H1 . Observe that, by (2.8), for λ > 0,
1
i
ZΦλj,k (z, t) = ∂z − z̄∂t eiλt ϕj,k (λ 2 z)
4
1
1
1
1
iλt
2
2
2
=e
λ (∂z ϕj,k )(λ z) + λz̄ϕj,k (λ z)
4
1
1
z̄
1
1
1
iλt
2
2
2
2
=e
λ Zϕj,k (λ z) − λ ϕj,k (λ z) + λz̄ϕj,k (λ z)
4
4
r
kλ λ
=−
Φ
(z, t) .
2 j,k−1
Integrating by parts, we find that
c (λ, j, k) =
Zf
Similarly,
r
kλ ˆ
f (λ, j, k − 1) .
2
r
(k + 1)λ ˆ
f (λ, j, k + 1) .
2
For λ < 0, the two expressions are interchanged:
r
r
(k
+
1)|λ|
c̄ (λ, j, k) = k|λ| fˆ(λ, j, k − 1) .
c (λ, j, k) = −
Zf
fˆ(λ, j, k + 1) ,
Zf
2
2
c̄ (λ, j, k) = −
Zf
Define the matrices

0
0

.
.
Uλ = 
.
0

..
.
p
···
|λ|/2 p0
|λ| · · ·
0
..
..
..
.
.
.
0
0
..
.

···
··· 


 ,

p

k|λ|/2

..
..
.
.
0
0
78
CHAPTER IV

0
···
p0
 − |λ|/2
0
···


p
..

.
0
− |λ|


..
..
Lλ = 
..
.
.
.



0
0

..
..
.
.
0
0
···
···
0
p
− kλ/2
..
.
..
.
Then, for λ > 0,
(3.3)
and, for λ < 0,
(3.4)
c (λ) = fˆ(λ)Uλ ,
Zf
c̄ (λ) = fˆ(λ)L ,
Zf
λ
c (λ) = fˆ(λ)Lλ ,
Zf
c̄ (λ) = fˆ(λ)U ,
Zf
λ






 .





In all these cases, we see that the action of a left-invariant differential operator
on f is reflected on the Fourier transform side by right multiplication by a matrix
depending on λ. The same holds for the integral operators T f = f ∗ k in (1.1). By
(2.13) we know in fact that
Tcf (λ) = fˆ(λ)k̂(λ) ,
at least for k ∈ L1 (H1 ). We prove now that this is a general fact for bounded
operators on L2 (H1 ).
Definition. We say that a matrix-valued funtion M (λ) = m(λ, j, k) j,k∈N is a
bounded Fourier multiplier for H1 if m(·, j, k) ∈ L∞ (R) for every j, k ∈ N, and if
kM k∞ = ess sup kM (λ)kL(`2 ) < ∞ .
Theorem 3.1. If M is a bounded Fourier multiplier for H1 , the requirement that
ˆ
T[
M f (λ) = f (λ)M (λ)
(3.5)
defines a bounded left-invariant operator TM on L2 (H1 ), with kTM k
L L2 (H1 )
=
kM k∞ .
Conversely, for any bounded left-invariant operator T on L2 (H1 ), there is a
bounded Fourier multiplier M such that T = TM .
Proof. Assume that M is a bounded Fourier multiplier. Since, for any pair of
operators A, B on a Hilbert space H,
kABkHS ≤ kAkHS kB||L(H) ,
taking f ∈ L2 (H1 ),
Z
Z
λ
2
2
kπ (f )M (λ)kHS |λ| dλ ≤ kM k∞ kπ λ (f )k2HS |λ| dλ .
R
R
HEISENBERG GROUP
By Plancherel’s formula, TM is bounded on L2 (H1 ), and kTM k
79
L L2 (H1 )
kM k∞ . The opposite inequality will follow from the second part of the proof.
By (2.11),
λ
ˆ
L\
(w,u) f (λ) = Φ (w, u)f(λ) .
Therefore
≤
\T f )(λ) = Φλ (w, u)Tcf(λ) = (L\
(L(w,u)
(w,u) f )(λ)k̂(λ) ,
which implies that T commutes with L(w,u) for every (w, u) ∈ H1 .
Suppose now that T commutes with left translations and is bounded on L2 (H1 ).
As a consequence of the Schwartz kernel theorem, as it has already been mentioned,
there is a distribution u ∈ S 0 (H1 ) such that T f = f ∗ u for f ∈ S(H1 ). We want
to show that the distributions ûj,k defined in (2.20) are in fact bounded functions,
and that M (λ) = {uj,k (λ)}j,k is a bounded Fourier multiplier.
Take f ∈ S0 (H1 ) such that fˆ(λ, j, k) = ψ(λ)δj,0 δk,p , where ψ ∈ D(R \ {0}) and
p ∈ N. Then, by Lemma 2.9,
\
(f
∗ u)j,k = δj,0 ψ ûp,k .
If f ∗ u ∈ L2 (H1 ), as we are assuming, a necessary condition is that for every
ψ as above and every p, k ∈ N, ψ ûp,k be square integrable in λ. This implies that
each ûj,k is locally integrable on R \ {0}. We can then define M (λ) for a.e. λ.
Take now an infinite matrix A with only finite many entries different from 0, and
ψ ∈ D(R \ {0}). Let f ∈ S0 (H1 ) be such that fˆ(λ) = ψ(λ)A. We have
Z
R
kfˆ(λ)M (λ)k2HS
Z
|ψ(λ)|2 kAM (λ)k2HS |λ| dλ
R
Z
2
kfˆ(λ)k2HS |λ| dλ
≤ kT kL(L2 (H1 ))
ZR
|ψ(λ)|2 kAk2HS |λ| dλ .
= kT k2L(L2 (H1 ))
|λ| dλ =
R
Since this must hold for every ψ, it follows that for a.e. λ,
kAM (λ)kHS ≤ kT kL(L2 (H1 )) kAkHS .
Given v ∈ `2 with only finitely many components different from 0, take Av as
the matrix having the components of v on the top row, and 0 on the others. Then
kAv M (λ)kHS = kM (λ)∗ vk`2 ≤ kT kL(L2 (H1 )) kAv kHS = kT kL(L2 (H1 )) kvk`2 .
Therefore kM (λ)kL(`2 ) = kM (λ)∗ kL(`2 ) ≤ kT k. That each ûj,k ∈ L∞ (R) follows
easily from the fact that it is measurable, and
ûj,k = hM (λ)ek |ej i`2 .
Hence M is a bounded Fourier multiplier and kM k∞ ≤ kT kL(L2 (H1 )) . 80
CHAPTER IV
How to transfer all this discussion to right-invariant operators is rather clear.
The right-invariant analogues of the Zj , Z̄j are
Z (r) =
i
1 (r)
(X − iY (r) ) = ∂z + z̄∂t ,
2
4
Z̄ (r) =
1 (r)
i
(X + iY (r) ) = ∂z̄ − z∂t .
2
4
\
(r) f (λ) can be expressed as in (3.3) and (3.4), only with the order of the
Then Z
two factors on the right-hand side reversed. This fact goes together with identities
like
r
(j + 1)λ λ
(3.6)
Z (r) Φλj,k (z, t) = −
Φj+1,k (z, t) ,
2
valid for λ > 0. Precisely, we have
(3.7)
\
(r) f (λ) = U fˆ(λ) ,
Z
λ
\
(r) f (λ) = L fˆ(λ) ,
Z̄
λ
\
(r) f (λ) = L fˆ(λ) ,
Z
λ
\
(r) f (λ) = U fˆ(λ) ,
Z̄
λ
and, for λ < 0,
(3.8)
Theorem 3.1 has the same formulation, except for the order of the two factors
in (3.5), for right-invariant convolution operators, T f = k ∗ f .
4. Radial functions and diagonal multipliers
In Section 1 we presented the rotations (z, t) 7−→ (U z, t) of Hn , with U a unitary
transformation of Hn . For a function f defined on Hn , we set
fU (z, t) = f (U z, t) .
The fact that rotations are automorphisms of Hn implies that
(4.1)
(f ∗ g)U = fU ∗ gU .
If n = 1, rotations are just scalar multiplications by complex numbers of modulus
1. We then write
fθ (z, t) = f (eiθ z, t) .
We say that a function on H1 is radial if it depends only on |z| and t, or,
equivalently, if fθ = f for every θ. More generally, we say that f is of type m ∈ Z if
fθ = eimθ f
for every θ, or, equivalently, if e−im arg z f (z, t) is radial.
These notions can be adapted to distributions as follows: a distribution u is of
type m if
hu, fθ i = e−imθ hu, f i ,
for every test function f .
Clearly, by expansion in Fourier serie in the angular variable, every function
(or distribution) decomposes as a sum of functions of the different types (with
convergence in a sense that depends on the function itself).
The function Φλj,k is of type k − j if λ > 0 and of type j − k if λ < 0. This gives
the following result.
HEISENBERG GROUP
81
Lemma 4.1. A function u is of type m if and only if fˆ(λ, j, k) = 0, unless λ > 0
and j − k = m, or λ < 0 and k − j = m. In particular, f is radial if and only if
fˆ(λ) is diagonal for every λ 6= 0.
If u is a radial tempered distributions, then ûj,k = 0 for j 6= k.
Proof. Suppose that f is of type m, and take λ > 0. Then
Z
imθ ˆ
e f (λ, j, k) =
f (eiθ z, t)Φλj,k (z, t) dz dt
H
Z 1
f (z, t)Φλj,k (e−iθ z, t) dz dt
=
=e
H1
−i(k−j)θ
fˆ(λ, j, k) ,
for every θ. So, if m 6= j − k, necessarily fˆ(λ, j, k) = 0. The rest of the proof follows
in the same way. Consider now a convolution operator T f = f ∗ u, with u. If u is of type m,
(T f )θ = fθ ∗ uθ = eimθ fθ ∗ u = eimθ T (fθ ) .
Conversely, if (T f )θ = eimθ T (fθ ) for every θ and every test function f , then
u ∗ f = eimθ (u ∗ fθ )−θ = eimθ u−θ ∗ f .
Hence u = eimθ u−θ , i.e. u is of type m.
The special case m = 0 concerns the left-invariant operators that also commute
with rotations.
Theorem 4.2. Let T f = f ∗ u be a bounded operator on L2 (H1 ). The following
conditions are equivalent:
(i) T commutes with rotations;
(ii) u is radial;
(iii) the Fourier multiplier M (λ) is diagonal for a.e. λ;
(iv) T = µ(iT, L), for some bounded spectral multiplier µ(λ, ξ) on the Heisenberg
fan F1 .
If these conditions are satisfied, then µ(λ, ξ) and the diagonal entries m(λ, k, k)
of M (λ) are related by the identity
m(λ, k, k) = µ λ, |λ|(2k + 1) .
Proof. The equivalence between (i) and (ii) follows from the previous remarks. The
implication (ii)⇒(iii) follows from Lemma 4.1, since ûj,k = m(·, j, k).
Given a Borel
subset ω in R2 , define the bounded Fourier multiplier Mω =
Mω (λ, j, k) j,k as
Mω (λ, j, k) =
1
0
if λ, |λ|(2k + 1) ∈ ω and j = k ,
otherwise ,
and let E(ω) be the corresponding orthogonal projection on L2 (H1 ). This define a
resolution of the identity and its support is the Heisenberg fan F1 .
82
CHAPTER IV
For f, g ∈ S(H1 ), it follows from the Plancherel formula that
Z
X
1
νf,g (ω) = hE(ω)f |gi =
fˆ(λ, j, k)ĝ(λ, j, k) |λ| dλ .
(2π)2 R (j,k):(λ,|λ|(2k+1))∈ω
This implies that
Z
ϕ(λ, ξ) dνf,g (λ, ξ) =
F1
Z X
(4.2)
1
ϕ
λ,
|λ|(2k
+
1)
fˆ(λ, j, k)ĝ(λ, j, k) |λ| dλ .
=
(2π)2 R
j,k∈N
Let
A=
Z
λ dE(λ, ξ) ,
B=
F1
Z
ξ dE(λ, ξ) .
F1
The domain D(A) of A consists of the functions f such that
Z X
Z
1
2
λ2 |fˆ(λ, j, k)|2 |λ| dλ < ∞ .
λ dνf,f (λ, ξ) =
2
(2π)
R
F1
j,k∈N
Let V be the space of finite families v = {vj,k }(j,k)∈Bv of functions vj,k ∈ D(R \
{0}). By Corollary 2.9, V consists of the finite families {ĝ(·, j, k)} with g ∈ S 0 (H1 ).
Then
R P
Z X
12
ˆ
(j,k)∈Bv λf (λ, j, k)vj,k (λ) |λ| dλ
R
λ2 |fˆ(λ, j, k)|2 |λ| dλ
= sup 12
v∈V
R P
R j,k∈N
2
(j,k)∈Bv |vj,k (λ)| |λ| dλ
R
= 2π
= 2π
hf | − iT gi
kgk2
g∈S0 (H1 )
sup
hiT f |gi
g∈S0 (H1 ) kgk2
sup
Since S0 (H1 ) is dense in L2 (H1 ), f ∈ D(A) if and only if iT f (defined as a
distribution) is in L2 , i.e. D(A) = D(iT ). Moreover, for f, g ∈ S(H1 ),
Z
hAf |gi =
λ dνf,g (λ, ξ)
F1
Z X
1
=
λfˆ(λ, j, k)ĝ(λ, j, k) |λ| dλ
(2π)2 R
j,k∈N
= hiT f, gi .
Hence A = iT . A similar argument shows that B = L.
Applying (4.2) with ϕ = µ, we then have, by (2.16),
Z X
1
fˆ(λ, j, k)ĝ(λ, j, k) |λ| dλ
hµ(iT, L)f |gi =
µ
λ,
|λ|(2k
+
1)
(2π)2 R
j,k∈N
Z
1
ˆ(λ)M (λ)ĝ(λ)∗ |λ| dλ
f
tr
=
(2π)2 R
= hTM f |gi .
HEISENBERG GROUP
83
This proves the equivalence between (iii) and (iv).
Finally, assume that T = TM , with M satisfying (iii). Given f ∈ S(Hn ), let fm ,
m ∈ Z, be the components of f of type m. Then T fm is also of type m by Lemma
4.1. Therefore,
X
T (fθ ) =
T (fm )θ
m∈Z
X
=
eimθ T (fm )
m∈Z
X
=
T (fm )θ
m∈Z
= (T f )θ .
This shows that (iii)⇒(i). 5. Radiality in Hn
We extend the Fourier analysis on H1 presented in the last three sections to Hn .
A large part of what we are going to say is a straightforward adaptation of what
has been presented in detail for H1 (only with more complicated notation), and we
leave the verification to the reader. The new facts will arise when we will present
the different notions of radiality.
Let j = (j1 , . . . , jn ), k = (k1 , . . . , kn ) be two n-tuples in Nn . For λ ∈ R \ {0}, we
set
Φλj,k (z, t) = eiλt
(5.1)
n
Y
ϕλji ,ki (zi ) .
i=1
For f ∈ L1 (Hn ), define
fˆ(λ, j, k) =
(5.2)
and, with Φλ (z, t) = Φλj,k (z, t)
(5.3)
Z
Hn
j,k∈Nn
fˆ(λ) = fˆ(λ, j, k)
f (z, t)Φλj,k (z, t) dz dt ,
,
j,k∈Nn
=
Z
f (z, t)Φλ (z, t) dz dt .
Hn
Theorem 5.1.
(1) Φλ (z, t) is unitary
for every (z, t);
(2) Φλ (z, t), (w, u) = Φλ (z, t)Φλ (w, u), Φλ (z, t)−1 = Φλ (z, t)∗ ;
(3) LΦλj,k = |λ| 2|k| + n Φλj,k and T Φλj,k = iλΦλj,k ;
(4) f[
∗ g(λ) = fˆ(λ)ĝ(λ) for every f, g ∈ L1 (Hn ) and every λ 6= 0;
c (λ, j, k) = |λ| 2|k| + n fˆ(λ, j, k) and Tcf (λ, j, k) = −iλfˆ(λ, j, k);
(5) Lf
84
CHAPTER IV
(6) if λ > 0 and ei = (0, 0, · · · , 1, . . . , 0), with the 1 in the i-th position,
r
ki λ ˆ
f (λ, j, k − ei ) ,
2
r
(ki + 1)λ ˆ
d̄
Z
f (λ, j, k + ei ) ,
i f (λ, j, k) = −
2
r
(ji + 1)λ ˆ
\
(r)
f (λ, j + ei , k) ,
Zi f (λ, j, k) =
2
r
ji λ ˆ
\
(r)
f (λ, j − ei , k) ,
Z̄i f (λ, j, k) = −
2
d
Z
i f (λ, j, k) =
(7) if λ < 0,
r
(ki + 1)|λ| ˆ
d
f (λ, j, k + ei ) ,
Z
i f (λ, j, k) = −
2
r
ki |λ| ˆ
d̄
Z
f (λ, j, k − ei ) ,
i f (λ, j, k) =
2
r
ji |λ| ˆ
\
(r)
Zi f (λ, j, k) = −
f (λ, j − ei , k) ,
2
r
(ji + 1)|λ| ˆ
\
(r)
f (λ, j + ei , k) ,
Z̄i f (λ, j, k) =
2
(8) the following Plancherel formula holds, for f ∈ L2 (Hn ):
Z
1
|f (z, t)| dz dt =
(2π)n+1
Hn
2
Z
R
kfˆ(λ)k2HS |λ|n dλ .
(9) the following inversion formula holds, for f ∈ S(Hn ):
1
f (z, t) =
(2π)n+1
Z
R
tr fˆ(λ)Φλ (z, t)∗ |λ|n dλ .
Moreover, Theorem 3.1 has the same formulation on Hn .
The formulas at points (6) and (7) can be expressed in analogy with (3.3) and
(3.4). Define the matrices Uλ,i , Lλ,i with indices (j, k) ∈ (Nn )2 by
(5.4)
(Uλ,i )j,k =
r
ki |λ|
δj,k−ei ,
2
(Lλ,i )j,k = −
r
(ki + 1)|λ|
δj,k+ei .
2
Then, for λ > 0,
(5.5)
and, for λ < 0,
(5.6)
d
ˆ
Z
i f (λ) = f (λ)Uλ,i ,
d̄
ˆ
Z
i f (λ) = f (λ)Lλ,i ,
d
ˆ
Z
i f (λ) = f (λ)Lλ,i ,
d̄
ˆ
Z
i f (λ) = f (λ)Uλ,i ,
HEISENBERG GROUP
85
The representation-theoretic formulation of (1) and (2) can be given in terms of
the unitary representations π λ (for λ 6= 0) of Hn on `2 (Nn ), such that π λ (z, t) is
the operator defined by the matrix Φλ (z, t) in the canonical basis.
If n > 1, we must distinguish between two notions of radiality.
We say that a function f (z, t) is radial if it only depends on |z| and t. This is
equivalent to saying that f (U z, t) = f (z, t) for every unitary n × n matrix U and
every (z, t).
We say that a function f (z, t) is poliradial if it depends on |z1 |, . . . , |zn | and t.
This is equivalent to saying that f (Uθ z, t) = f (z, t) for every unitary diagonal n × n
matrix,

 iθ1
e
0
···
0
 0
eiθ2 · · ·
0 

,
..
.. 
Uθ =  ..
..
.
.
.
. 
0
0
···
eiθn
and every (z, t). We shall write fθ (z, t) for f (Uθ z, t).
The natural extensions of Lemma 4.1 and Theorem 4.2 concerns poliradial functions and joint spectral multipliers of T and the partial sub-Laplacians Li = −Xi2 −
Yi2 . We group them in one statement, disregarding the first part of Lemma 4.119 .
Theorem 5.2. A function f ∈ L1 (Hn ) is poliradial if and only if fˆ(λ) is diagonal
for every λ 6= 0. For a bounded operator T f = f ∗ u on L2 (H1 ), the following
conditions are equivalent:
(i) T commutes with the rotations Uθ , i.e. T (fθ ) = (T f )θ for every θ;
(ii) u is poliradial;
(iii) the Fourier multiplier M (λ) is diagonal for a.e. λ;
(iv) T = µ(iT, L1 , . . . , Ln ), for some bounded spectral multiplier µ(λ, ξ1 , . . . , ξn ).
If these conditions are satisfied, then µ(λ, ξ1 , . . . , ξn ) and the diagonal entries
m(λ, k, k) of M (λ) are related by the identity
m(λ, k, k) = µ λ, |λ|(2k1 + 1), . . . , |λ|(2kn + 1) .
The condition on fˆ(λ) characterizing radial functions must be more restrictive.
We shall show that it consists in the fact that fˆ(λ, k, k) = fˆ(λ, k0 , k0 ) if |k| = |k0 |.
Lemma 5.3. A polyradial function f ∈ S(Hn ) is radial if and only if for every
i, i0 ,
(r)
(r)
Z i Z i0 f = Z i0 Z i f .
Proof. Write f (z, t) = g(r1 , . . . , rn , t), with ri = |zi |2 . Then f is radial if and only
if g only depends on r1 + · · · + rn and t, i.e. if and only if ∂ri g = ∂ri0 g for every
i, i0 . Since
z̄i ∂zi0 f − z̄i0 ∂zi f = z̄i z̄i0 ∂ri0 g − z̄i0 z̄i ∂ri g = z̄i z̄i0 (∂ri0 − ∂ri )g ,
it follows that f is radial if and only if
(z̄i ∂zi0 − z̄i0 ∂zi )f = 0
19 One
can introduce the notion of type m, for m ∈ Zn , and extend that part too.
86
CHAPTER IV
for every i, i0 . Now,
i
i
(r)
(r)
(Zi Zi0 − Zi0 Zi )f = ∂zi − z̄i ∂t ∂zi0 + z̄i0 ∂t f
4
4
i
i
0
− ∂zi0 − z̄i ∂t ∂zi + z̄i ∂t f
4
4
i
= (z̄i0 ∂zi − z̄i ∂zi0 )∂t f .
2
(r)
(r)
(r)
Therefore, if f is radial, (Zi Zi0 − Zi0 Zi )f = 0. Conversely, if Zi Zi0 f =
(r)
Zi0 Zi f , we obtain that ∂t f is radial. Since ∂t f ∈ S(Hn ),
f (z, t) =
Z
t
∂t f (z, u) du
−∞
is also radial. Theorem 5.4. A function f ∈ S(Hn ) is radial if and only if fˆ(λ) is diagonal and
fˆ(λ, k, k) only depends on |k|.
Proof. Assume that f is radial. In particular, it is poliradial, hence fˆ(λ) is diagonal.
(r)
(r)
Moreover, by Lemma 5.3, Zi Zi0 f = Zi0 Zi f for every i, i0 . By Theorem 5.1, (6)
and (7), we obtain that, for λ > 0,
\
(r)
(Zi Zi0 f )(λ, k − ei0 , k + ei ) =
and
\
(r)
(Zi0 Zi f )(λ, k0 − ei , k0 + ei0 ) =
p
(ki + 1)ki0 λ ˆ
f (λ, k, k) ,
2
p 0 0
ki (ki0 + 1)λ ˆ
f (λ, k0 , k0 ) .
2
Fix two different indices i, i0 and take k, k0 such that k + ei = k0 + ei0 . Then the
two left-hand sides coincide, and so do the expressions under square root. We must
then have
fˆ(λ, k, k) = fˆ(λ, k0 , k0 ) .
This means that, moving one unit from one entry of k to another entry, the value
fo the Fourier coefficient does not change. Repeating this operation, we can pass
from any k to any other k0 with the length.
For λ < 0 the argument is the same, and this proves the first part of the statement.
Suppose conversely that f ∈ S(Hn ) and fˆ(λ, j, k) = δj,k ν(λ, |k|). By the inversion formula,
1
f (z, t) =
(2π)n+1
Z X
R k∈N
with
Ψλk (z, t) =
ν(λ, k)Ψλk (z, t) |λ|n dλ ,
X
|k|=k
Φλk,k (z, t) .
HEISENBERG GROUP
87
Since each Φλk,k is poliradial, Ψλk is poliradial too. As in Section 3, for λ > 0,
r
ki λ λ
λ
Zi Φk,k = −
Φ
2 k,k−ei
and
(r)
Zi Φλk,k
Therefore
(r)
Zi Zi0 Φλk,k
=−
r
(ki + 1)λ λ
Φk+ei ,k .
2
p
ki (ki0 + 1)λ λ
Φk+ei0 ,k−ei .
=
2
(r)
(r)
Summing over k and setting k0 = k+ei0 −ei , we find that Zi Zi0 Ψλk = Zi0 Zi Φλk .
As in the proof of Lemma 5.3, this implies that (z̄i0 ∂zi − z̄i ∂zi0 )∂t Ψλk = 0. But, since
Ψλk (z, t) = ψkλ (z)eiλt , we conclude that ψkλ only depends on |z|.
Finally, repeating the same argument for λ < 0, we conclude that Ψλk is radial
for every λ 6= 0, and hence f is radial. We need at this point to describe the orthogonal projection P from L2 (Hn ) onto
the subspace of radial functions. This requires some notions concerning the group
U (n) of unitary transformations of Cn . With the natural topology, induced from
2
Cn , U (n) is compact and the proup operations are continuous. The basic fact is
the existence of a unique Borel probability measure m (called the Haar measure)
which is invariant under left and right translations20 , i.e. such that
m(E) = m(gE) = m(Eg) = m(E −1 ) ,
for every Borel set E and every g ∈ U (n). As a consequence,
Z
Z
f (x) dm(x) =
f (gx) dm(x)
U (n)
U (n)
Z
Z
=
f (xg) dm(x) =
f (x−1 ) dm(x) ,
U (n)
U (n)
for every integrable function f and every g ∈ U (n).
Lemma 5.5. The orthogonal projection P from L2 (Hn ) onto the subspace of radial
functions L2rad (Hn ) is given by
Z
(5.7)
P f (z, t) =
f (U z, t) dm(U ) ,
U (n)
and
(5.8)
Pcf (λ, j, k) = δj,k
|k| + n − 1
n−1
−1
X
fˆ(λ, k0 , k0 ) .
k0 :|k0 |=|k|
P is well-defined and continuous on the following spaces:
(i) Lp (Hn ), for 1 ≤ p ≤ ∞;
(ii) S(Hn ) into itself continuously, and it can therefore be extended by duality
to S 0 (Hn );
(iii) S0 (Hn ), defined as in Corollary 2.9.
20 See,
e.g., the notes of the course Analisi di Fourier non commutativa.
88
CHAPTER IV
If X denotes each of these spaces, the image of P in X is the subspace X rad of
all radial functions (or distributions) in the space itself. In particular S rad (Hn ) and
S0,rad (Hn ) are dense in Lprad (Hn ) for 1 ≤ p < ∞.
Proof. Since
P f (U z, t) =
Z
0
0
f (U U z, t) dm(U ) =
U (n)
Z
f (U 0 z, t) dm(U 0 ) = P f (z, t) ,
U (n)
the image of P consists of radial functions. If f is already radial, it is clear that
P f = f . Hence P 2 = P and the image consists of all radial functions. Given
f, g ∈ L2 (Hn ),
Z Z
hP f |gi =
f (U z, t)g(z, t) dm(U ) dz dt
Hn U (n)
Z
Z
=
f (U z, t)g(z, t) dz dt dm(U )
U (n) Hn
Z
Z
=
f (z, t)g(U −1 z, t) dz dt dm(U )
U (n)
Hn
= hf |P gi .
Hence P is an orthogonal projection.
Formula (5.8) is a direct consequence of Theorem 5.4 and the Plancherel formula,
once we have observed that the binomial coefficient in front of the sum gives the
number of k0 with the same length of k..
If f ∈ Lp (Hn ), the Minkowski integral inequality gives
Z
Z
kP f kp = fU dm(U ) ≤
kfU kp dm(U ) = kf kp .
U (n)
p
U (n)
Boundedness of P on S(Hn ) follows from (5.7) by differentiation under the integral sign, and on S0 (Hn ) from (5.8).
The last part is then obvious. Theorem 5.6. For a bounded operator T f = f ∗ u on L2 (H1 ), the following conditions are equivalent:
(i) T commutes with all the rotations of Hn , i.e. T (fU ) = (T f )U for every
U ∈ U (n);
(ii) T maps radial functions into radial functions;
(iii) u is radial;
(iv) the Fourier multiplier M (λ) has the form
M (λ) = δj,k ν λ, |k|
;
j,k
(v) T = µ(iT, L), for some bounded spectral multiplier µ(λ, ξ) on Fn .
If these conditions are satisfied, then the spectral multiplier µ(λ, ξ) and the diagonal entries ν(λ, k) of M (λ) are related by the identity
ν(λ, k) = µ λ, |λ|(2k + n) .
HEISENBERG GROUP
89
Proof. The implication (i)⇒(ii) is trivial, because if f is radial, (T f )U = T (fU ) =
T f for every U , hence
T f is radial. To prove that (ii)⇒(iii), take a radial function
R
ϕ ∈ S(Hn ) with Hn ϕ = 1. Then the functions
z
t ϕε (z, t) = 2n+2 ϕ , 2
ε
ε ε
1
form an approximate identity as ε → 0. In particular,
lim T (ϕε ) = lim u ∗ ϕε = u
ε→0
ε→0
in S 0 (Hn ). But T (ϕε ) is radial by assumption, hence u is also radial.
Assume now that u is radial. Then, for any U ∈ U (n),
(T f )U = (u ∗ f )U = uU ∗ fU = u ∗ fU = T (fU ) ,
which give the implication (iii)⇒(i).
The implication (iv)⇒(ii) is obvious by Theorem 5.4. To prove that (i)⇒(iv),
we fix ϕ ∈ D(R \ {0}) and k ∈ N, and define
1
ϕ̃k (z, t) =
(2π)n+1
Z
ϕ(λ)
R
X
|k|=k
Φλk,k (z, t) |λ|n dλ .
Then ϕ̃k ∈ S0 (Hn ) and, by Theorem 5.4, it is radial. Hence, since we know that
(i) implies (ii), T ϕ̃k is also radial.
Since T commutes with diagonal rotations, we know from Theorem 5.2 that the
Fourier multiplier of T is diagonal. Let m(λ, k, k) be its entries on the diagonal.
Then
c̃k (λ, k, k)m(λ, k, k) = δ|k|,k ϕ(λ)m(λ, k, k) ,
Td
ϕ̃k (λ, k, k) = ϕ
must depend only on λ and |k|. By the arbitrarity of ϕ and k, m(λ, k, k) = ν(λ, |k|).
Finally the equivalence between (iv) and (v) is proved as in Theorem 4.2. Define
(5.9)
Ψλk (z, t)
=
k+n−1
n−1
−1 X
Φλk,k (z, t) ,
|k|=k
for k ∈ N. It follows from this definition that
(5.10)
Ψλk (z, t) = eiλt ψk |λ||z|2 ,
where ψk is a Schwartz function.
The relevance of the functions Ψλk in our context is clarified by the next statement.
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CHAPTER IV
Proposition 5.7. Up to scalar multiples, Ψλk is the unique radial function in
span {Φλk,k : |k| = k}. If |k| = k, then P Φλk,k = Ψλk , and, if f ∈ L1 (Hn ),
(5.11)
Pcf (λ, k, k) =
Z
Hn
f (z, t)Ψλk (z, t) dz dt .
Proof. We start by proving (5.10). If f ∈ L1 ∩ L2 (Hn ), it follows directly from
(5.8), and it extends to any integrable f by continuity, since Ψλk ∈ L∞ (Hn ).
Take now
X
u=
ck Φλk,k ∈ L∞ (Hn ) .
|k|=k
Then, for f ∈ L1 (Hn ),
hP u, f i = hu, P f i
X
=
ck Pcf (λ, k, k)
|k|=k
=
X
|k|=k
ck hΨλk , f i ,
by (5.11). Hence P u is a scalar multiple of Ψλk , and the rest of the statement follows
easily. 6. Applications
Fourier analysis on Hn can be used to derive the regularity properties of the
sub-Laplacian. We shall prove estimates showing that if f and Lk f are in L2 (Hn ),
then all the derivatives of f up to a certain order are (at least locally) in L2 . The
general name for this type of results is sub-elliptic estimates, and they are typical
of hypoelliptic operators.
In the second part of this Section, we characterize Fourier transforms of radial
Schwartz functions.
The most efficient way to state hypoelliptic estimates for L is in terms of leftinvariant vector fields, because they can be stated in global form.
Consider “higher-order left-invariant derivatives” of f , meant as expressions like
T 2 X1 Y2 Y1 X23 f , or like Z̄22 Z1 Z̄1 T f . The order of the factors in these expressions is
relevant; we call them non-commutative monomials 21 . Clearly one can switch from
non-commutative polynomials in the Xj , Yj , T to non-commutative polynomials in
the Zj , Z̄j , T by simple formal manipulations. The use of the Zj , Z̄j is preferable
for us, because formulas for the Fourier transform are simpler. Observe also that,
due to the relations
i
[Xj , Yj ] = T ,
[Zj , Z̄j ] = T ,
2
21 We
mention that it follows from the Poincaré-Birckhoff-Witt theorem (see, e.g., the notes
Sub-Laplacians on nilpotent Lie groups). that every left-invariant differential operator on H n can
be written as a non-commutative polynomial.
HEISENBERG GROUP
91
different non-commutative polynomials can give the same differential operator22 .
In particular, one can always replace a non-commutative polynomial by another
one not containing T , without altering the differential operator.
Let P (Z, Z̄) be a non-commutative polynomial in the Zj , Z̄j only. The degree
of P is defined in the usual way. If T also appears in P , it must be counted
as a factor of degree 2. With this convention, the non-isotropic order of a leftinvariant diffeerential operator is well defined, as the degree of any non-commutative
polynomial representing it.
Theorem 6.1. Let N ∈ N, and assume that f and LN/2 f are in L2 (Hn ). Then
also P (Z, Z̄, T )f ∈ L2 (Hn ) for every P of non-isotropic degree at most N , and
kP f k2 ≤ CN kf k2 + kLN/2 f k2 .
Moreover, all partial derivatives of f up to the order [N/2] are locally in L 2 .
Proof. It is sufficient to take a monomial P = P (Z, Z̄) in the Zj , Z̄j of degree
d ≤ 2N . For each λ 6= 0, Pcf (λ) is given by
Pcf (λ) = fˆ(λ)P (Uλ , Lλ ) ,
or
Pcf (λ) = fˆ(λ)P (Lλ , Uλ ) ,
depending on the signum of λ, where Uλ , Lλ stand for the matrices Uλ,i , Lλ,i in
(5.4). The diagonal matrix Dλ with
N/2
(Dλ )k,k = 1 + |λ|(2|k| + n)
is such that
Therefore, if
(I +\
LN/2 )f (λ) = fˆ(λ)Dλ .
Mλ =
we have that
P (Uλ , Lλ )Dλ−1
P (Lλ , Uλ )Dλ−1
if λ > 0 ,
if λ < 0 ,
Pcf (λ) = (I +\
LN/2 )f (λ)Mλ .
Observe that both P (Uλ , Lλ ) and P (Lλ , Uλ ) have non-zero entries only on one
single diagonal, and the k-th entry along this diagonal is dominated by |λ|(|k| +
d/2
1)
. Therefore the norms kMλ kL(`2 ) are uniformly bounded in λ. It follows from
Theorem 3.1 (which holds also on Hn , as already mentioned in Section 5) that
kP f k2 ≤ Ckf + LN/2 f k2 ≤ C kf k2 + kLN/2 f k2 .
The last part of the statement follows from the fact that, by the explicit expression of the vector fields, L2 -estimates for Zi g, Z̄i g, T g imply local L2 -estimates
for ∂zi g, ∂z̄i g, ∂t g. An induction argument shows that, in our hypotheses, we can
control locally all partial derivatives of f up to order [N/2]. A similar argument, made simpler by the fact that all the matrices involved are
diagonal, gives the following result.
22 In
more correct terms, a non-commutative polynomial is an element of the tensor algebra
T over C generated by the Zj , Z̄j , T . The map assigning to each element of T the corresponding
composition of left-invariant vector fields on Hn has a kernel, equal to the ideal I generated by
the elements Zj ⊗ Z̄k − Z̄k ⊗ Zj − 2i δj,k T . The quotient T /I identifies left-invariant differential
operators on Hn and is called the universal enveloping algebra of the Lie algebra hn .
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CHAPTER IV
Theorem 6.2. . Assume that f and Ls f are in L2 (Hn ) for some s ∈ R+ . Then
Ls1 (i−1 T )s2 −s1 f ∈ L2 (Hn ), for s1 , s2 ≥ 0, s2 ≤ s, and
kLs1 (i−1 T )s2 −s1 f k2 ≤ Cs kf k2 + kLs f k2 .
A further refinement gives global Sobolev estimates for poliradial functions.
Corollary 6.3. Suppose that f is poliradial and that f, LN/2 f ∈ L2 (Hn ). Then
∂zα ∂z̄β ∂tm f ∈ L2 (Hn ) for all multi-indices α, β and all m ∈ N such that |α| + |β| +
2m ≤ N , and
k∂zα ∂z̄β ∂tm f k2 ≤ CN kf k2 + kLN/2 f k2 .
Proof. Consider the right-invariant sub-Laplacian
L
(r)
=−
n
X
(r) 2 (r)
(Xi )2 + (Yi
i=1
)
.
c (λ) = fˆ(λ)M (λ), then \
If Lf
L(r) f (λ) = M (λ)fˆ(λ). Since M (λ) is diagonal, it
commutes with fˆ(λ) if f is poliradial. Hence L(r) f = Lf . Since the analogue of
(r)
(r)
Theorem 6.1 also holds for right-invariant operators, we have that P (Zi , Z̄i )f ∈
L2 (Hn ) for every non-commutative monomial P of degree at most 2N .
Starting from the identities
(r)
Zi + Z i
= 2∂zi ,
(r)
Z̄i + Z̄i
= 2∂z̄i ,
and expressing ∂t as a commutator, we reduce the problem of estimating ∂zα ∂z̄β ∂tm f
to proving that
kQ(Z, Z̄, Z (r) , Z̄ (r) )f k2 ≤ C kf k2 + kLN/2 f k2 ,
if Q = Q(Z, Z̄, Z (r) , Z̄ (r) ) is a non-commutative monomial of degree at most N .
It is a general fact (and it can be easily verified from the explicit formulas or
from the Fourier transforms) that right-invariant vector fields commute with leftinvariant ones. Hence they also commute with L, and, by Theorem 3.1 extended
to Hn , with its spectral projection, and ultimately with its fractional powers. The
same can be said interchanging left and right.
We can therefore write
Q = Q1 (Z, Z̄)Q2 (Z (r) Z̄ (r) ) ,
where Q1 and Q2 are monomials of degrees d1 and d2 respectively, with d1 +d2 ≤ N .
By Theorems 6.1 and 6.2,
kQf k2 ≤ C kQ2 (Z (r) Z̄ (r) )f k2 + kLd1 /2 Q2 (Z (r) Z̄ (r) )f k2
≤ C kf k2 + k(L(r) )d2 /2 f k2 + kQ2 (Z (r) Z̄ (r) )Ld1 /2 f k2
≤ C kf k2 + k(L(r) )d2 /2 f k2 + kLd1 /2 f k2 + k(L(r) )d2 /2 Ld1 /2 f k2
= C kf k2 + kLd2 /2 f k2 + kLd1 /2 f k2 + kL(d1 +d2 )/2 f k2
≤ C kf k2 + kLN/2 f k2 .
HEISENBERG GROUP
93
We conclude this section with the proof of some identities providing conditions
on the Fourier transform side that correspond to decay at infinity of the function.
In Rn this is provided by formulas like F (xj f )(ξ) = i∂ξj fˆ(ξ). In Hn , staying within
[
2 f (λ) and with tf
b (λ).
radial functions, we look for formulas relating fˆ(λ) with |z|
If f is radial, we set
f˜(λ, k) = fˆ(λ, k)
if |k| = k.
Lemma 6.4. Assume that f ∈ S(Hn ) is radial. Then
and
2
^
2 f )(λ, k) =
(|z|
(2k + n)f˜(λ, k) − (k + n)f˜(λ, k + 1) − k f˜(λ, k − 1) ,
|λ|
g)(λ, k) = −i∂λ f˜(λ, k) − i nf˜(λ, k) − (k + n)f˜(λ, k + 1) + k f˜(λ, k − 1) .
(tf
2λ
Proof. Take f ∈ S(Hn ), not necessarily radial. For each i = 1, . . . , n,
(r)
Zi − Z i
i
= − z̄i ∂t ,
2
(r)
Z̄i − Z̄i
=
i
zi ∂ t .
2
Therefore
(z̄\
i ∂t f )(λ, j, k) =

p
√
 i 2ki λfˆ(λ, j, k − ei ) − i 2(ji + 1)λfˆ(λ, j + ei , k)
and

−i
p
p
2(ki + 1)|λ|fˆ(λ, j, k + ei ) + i 2ji |λ|fˆ(λ, j − ei , k)
(z\
i ∂t f )(λ, j, k) =

p
√
 i 2(ki + 1)λfˆ(λ, j, k + ei ) − i 2ji λfˆ(λ, j − ei , k)

−i
p
p
2ki |λ|fˆ(λ, j, k − ei ) + i 2(ji + 1)|λ|fˆ(λ, j + ei , k)
if λ > 0 ,
if λ < 0 ,
if λ > 0 ,
if λ < 0 .
Therefore, for every λ 6= 0,
2 2
ˆ
(|z\
i | ∂t f )(λ, j, k) = −2(ji + ki + 1)|λ|f(λ, j, k)
p
+ 2|λ| (ji + 1)(ki + 1)fˆ(λ, j + ei , k + ei )
p
+ 2|λ| ji ki fˆ(λ, j − ei , k − ei ) .
Summing over i and restricting to f radial and j = k, we obtain that
^
2 ∂ 2 f )(λ, k) = −2(2k + n)|λ|f˜(λ, k) + 2|λ|(k + n)f˜(λ, k + 1)
(|z|
t
˜ k − 1) .
+ 2|λ|k f(λ,
94
CHAPTER IV
2 ^
2
^
2 ∂ 2 f ) = (∂^
2
2
But (|z|
t
t |z| f ) = −λ (|z| f ), and this proves the first formula.
In order to prove the second formula, we start from the derivative of f˜(λ, k) in λ.
By (5.10) and (5.11), if f is radial,
f˜(λ, k) =
Z
Hn
f (z, t)eiλt ψk |λ||z|2 dz dt ,
so that
Z
tf (z, t)eiλt ψk |λ||z|2 dz dt
Hn
Z
+ sgn λ
|z|2 f (z, t)eiλt ψk0 |λ||z|2 dz dt
Hn
Z
g
= i(tf )(λ, k) + sgn λ
|z|2 f (z, t)eiλt ψk0 |λ||z|2 dz dt .
∂λ f˜(λ, k) = i
Hn
Observing that
n
X
i=1
zi ∂zi ψk |λ||z|2 = |λ||z|2 ψk0 |λ||z|2 ,
we obtain
(6.1)
g)(λ, k)
∂λ f˜(λ, k) = i(tf
Z X
n
1
−
∂zi zi f (z, t) Ψλk (z, t) dz dt
λ Hn i=1
g)(λ, k) − 1
= i(tf
λ
X
n ^
∂zi (zi f ) (λ, k)
i=1
P
(observe that
∂zi zi f ) is also radial).
(r)
(r)
Using again the fact that Z̄i − Z̄i = 2i zi ∂t , and that Zi + Zi = 2∂zi , we have
(r)
(r)
i∂t ∂zi (zi f ) = (Zi + Zi )(Z̄i − Z̄i )f .
With computations similar to the previous ones, we find that, for λ 6= 0,
λ ˆ
λ∂z\
(zi f )(λ, k, k) =
f (λ, k, k)+ki fˆ(λ, k−ei , k−ei )−(ki +1)fˆ(λ, k+ei , k+ei ) .
i
2
Summing over i,
X
n ^
1
∂zi (zi f ) (λ, k) = nf˜(λ, k) + k f˜(λ, k − 1) − (k + n)f˜(λ, k + 1) .
2
i=1
Inserting this identity in (6.1), we find the stated formula.
HEISENBERG GROUP
95
The formula indicated in Lemma 6.4 are rather complicated, since they involve
strange second-order differences in k. A considerable simplification occurs if we
2
combine the two formulas to express the Fourier transform of |z|4 ± it f . Setting
w± (z, t) =
(6.2)
and
(6.3)
|z|2
4
± it, we have the following identities:

k+n ˜

f (λ, k + 1) − f˜(λ, k)
 ∂λ f˜(λ, k) −
λ
^
(w
+ f )(λ, k) =

k ˜

f (λ, k) − f˜(λ, k − 1)
∂λ f˜(λ, k) −
λ



k ˜
f (λ, k) − f˜(λ, k − 1)
λ
^
(w
− f )(λ, k) =

 −∂ f˜(λ, k) + k + n f˜(λ, k + 1) − f˜(λ, k)
λ
λ
−∂λ f˜(λ, k) +
if λ > 0 ,
if λ < 0 ,
if λ > 0 ,
if λ < 0 .
96
CHAPTER IV
CHAPTER V
SPECTRAL MULTIPLIERS OF THE SUB-LAPLACIAN
1. The heat kernel on Hn
Our discussion of spectral Lp -multipliers of L on Hn begins with the study of
the heat kernel. This is defined as the kernel ps (z, t), for s > 0, such that
f ∗ ps = e−sL f .
We recall some general facts about semigroups and evolution equations on Lie
groups.
The semigroup property e−(s1 +s2 )L = e−s1 L e−s2 L implies that
ps1 +s2 = ps1 ∗ ps2
for every s1 , s2 > 0. This identity extends to s = 0 if we set p0 = δ0 , consistently
with the fact that e0L = I. Moreover, the map s 7−→ e−sL is continuous from
[0, ∞) to L(L2 ) with the strong topology.
d −sL
The identity ds
e
f = Le−sL f holds for every s > 0 and f ∈ L2 (Hn ). Therefore, the function u(s, z, t) = e−sL f (z, t) satisfies the homogeneous heat equation
(∂s + L)u = 0 ,
which implies that
(∂s + L)ps = 0
in the sense of distributions.
By the already cited Hörmander’s theorem23 , the operator ∂s + L is hypoelliptic,
which implies that ps (z, t) is smooth in (s, z, t).
Finally, it follows from Hunt’s theorem24 that the ps define probability measures,
i.e.
Z
(1.1)
ps (z, t) ≥ 0 ,
ps (z, t) dz dt = 1 .
Hn
We shall prove now further properties of the ps , using the Fourier analysis on
Hn .
23 See
L. Hörmander, Hypoelliptic second-order differential equations, Acta Math. vol.119
(1967), p.147-171.
24 See A. Hunt, Semigroups of measures on Lie groups, Trans. Amer. Math. Soc. vol.81
(1956), p.264-293.
Typeset by AMS-TEX
SPECTRAL MULTIPLIERS
97
By Theorem 5.6 in Chapter IV,
pes (λ, k) = e−s|λ|(2k+n) ,
so that, by the inversion formula in Theorem 5.1 of Chapter IV,
Z X
k + n − 1 −s|λ|(2k+n) λ
1
(1.2)
ps (z, t) =
e
Ψk (z, t) |λ|n dλ .
n+1
(2π)
n−1
R
k∈N
We show that ps is a Schwartz function, and, more precisely that it decays
exponentially at infinity.
Proposition 1.1. For λ 6= 0,
X k + n − 1
|λ||z|2
2n eiλt
e−s|λ|(2k+n) Ψλk (z, t) =
e− tanh(|λ|s) .
n
n−1
sinh (|λ|s)
k∈N
The proof is based on the following lemma.
Lemma 1.2. For 0 ≤ r < 1 and z ∈ C,
X
r k ϕk,k (z) =
k∈N
1+r
1 − 4(1−r)
|z|2
e
.
1−r
Proof. By (3.3) and (3.4) in Chapter I,
hj,k (z) = (−2)j e
|z|2
4
∂z̄k ∂zj e−
|z|2
2
,
and, by (2.2) in Chapter IV,
ϕk,k (z) =
Then
X
|z|2
(−2)k |z|2 k k − |z|2
1 |z|2
e 4 ∂z̄ ∂z e 2 = k e 4 ∆k e− 2 .
k!
2 k!
r k ϕk,k (z) = e
|z|2
4
k∈N
X rk
|z|2
|z|2
k − 2
4 F (r, z) ,
∆
e
=
e
2k k!
k∈N
where the series converges for r < 1 because |ϕj,k (z)| ≤ 1. Recalling that, for s > 0,
(1.3)
F(
2
1 − |z|2
e 4s ) = e−s|ζ| ,
2πs
we have
F (∆k e−
|z|2
2
) = π|ζ|2k e−
|ζ|2
2
.
Therefore
F (r, z) = πF −1
X rk
|ζ|2
2k − 2
|ζ|
e
2k k!
k∈N
1−r
2
= πF (e− 2 |ζ| )
|z|2
1 − 2(1−r)
.
=
e
1−r
−1
98
CHAPTER V
Multiplying by e
|z|2
4
, the proof is completed.
Proof of Proposition 1.1. Assume first that λ = 1. By (5.9) and (5.10) in Chapter IV, and by Lemma 1.2,
X k + n − 1
k∈N
n−1
e−s(2k+n) Ψ1k (z, t) = eit
=e
it
= eit
n X
Y
i=1
n
Y
X
e−s(2|k|+n) ϕk,k (z)
k∈Nn
e
−s(2k+1)
ϕk,k (zi )
k∈N
−2s
e−s
− 1+e −2s |zi |2
1−e
e
1 − e−2s
i=1
s
−s
eit
− es +e−s |z|2
e
−e
= s
e
(e − e−s )n
2n eit − |z|2
e tanh s .
=
sinhn s
1
For a generic λ, it is sufficient to replace t by λt, s by |λ|s, and z by λ 2 z or
1
|λ| 2 z̄, depending on the signum of λ. By (1.2), we then have
ps (z, t) =
(1.4)
1
2π n+1
1
=
2π n+1
Z
|λ||z|2
eiλt
− tanh(|λ|s)
|λ|n dλ
e
n
sinh
(|λ|s)
R
Z
λ|z|2
eiλt
e− tanh(λs) λn dλ .
n
R sinh (λs)
Observe that
(1.5)
1
ps (z, t) = s−(n+1) p1 s− 2 z, s−1 t ,
which is the analogue of the scaling property for the Gauss-Weierstrass heat kernel
for the Laplacian in Rn .
Corollary 1.3. For s > 0, ps ∈ S(Hn ).
Proof. We can take s = 1 and prove that Ft p1 is a Schwartz function. By (1.4),
Ft p1 (z, λ) =
λ|z|2
λn
− tanh λ
e
.
π n sinhn λ
This function is analytic in (z, λ) and all of its derivatives decay rapidly at
infinity. We introduce on Hn the homogeneous norm
(1.6)
1
1
(z, t) = |z|4 + 16t2 4 = |z|2 + 4it 2 = 2|w+ | 21 .
99
Proposition 1.4. The homogeneous norm (1.6) satisfies the inequality
(z, t)(w, u) ≤ (z, t) + (w, u) .
Proof. We have
(z, t)(w, u)2 = |z + w|2 + 4i t + u − 1 =mhz|wi 2
2
2
= |z| + |w| + 2hw|zi + 4i(t + u)
≤ |z|2 + 4it + |w|2 + 4iu + 2|z||w|
2
2 ≤ (z, t) + (w, u) + 2(z, t)(w, u)
2
= (z, t) + (w, u) .
The following estimate can be seen as a Gaussian estimate w.r. to the homogeneous norm (1.6).
Corollary 1.5. There is a > 0 such that
p1 (z, t) ≤ Ce
Proof. From (1.4) we obtain that
p1 (z, t) ≤
1
π n+1
Z
+∞
0
2
−a(z,t)
.
λ|z|2
λn
− tanh λ
dλ .
n e
sinh λ
Since the function λ/ tanh λ is bounded from below by 1, we have
2 Z +∞
2
λn
e−b|z|
dλ = Ce−b|z| .
(1.7)
p1 (z, t) ≤ n+1
n
π
sinh λ
0
Consider now
2
λ
1
− tanh
4λ |ζ|
.
n e
cosh λ
It can be extended analytically in λ to the strip λ+iτ : |τ | < π2 . For |τ | ≤ π2 −δ,
F p1 (ζ, λ + iτ ) is integrable in (ζ, λ) and rapidly decreasing in λ, uniformly in ζ and
τ . Therefore, a change of contour integration in the plane λ + iτ gives
Z
1
i t(λ+iτ )+<ehz|ζi
p1 (z, t) =
F p1 (ζ, λ + iτ )e
dζ dλ
(2π)2n+1 Cn ×R
Z
e−τ t
i tλ+<ehz|ζi
=
F p1 (ζ, λ + iτ )e
dζ dλ
(2π)2n+1 Cn ×R
Z
−τ t
F p1 (ζ, λ + iτ ) dζ dλ
≤ Ce
F p1 (ζ, λ) =
Cn ×R
≤ Cτ e
−τ t
.
Combining this with (1.7), we obtain that, for some α > 0,
−α |z|2 +|t|
p1 (z, t) ≤ Ce
.
2
This gives the conclusion, since (z, t) ∼ |z|2 + |t|. 100
CHAPTER V
2. Smooth multipliers and Schwartz kernels
In his Section, we use the estimates obtained for the heat kernel to prove that
certain spectral multipliers of L correspond to convolution with a Schwartz kernel.
Theorem 2.1. Let m be a smooth function supported on the interval 14 , 4 . Then
m(L)f = f ∗ k, where k ∈ S(Hn ).
This theorem will be proved in a few steps. The multiplier must be first decomposed appropriately, in order to take advantage of the heat kernel estimates.
1
Let µ(τ ) = m(− log τ ). Then µ is supported on [e−4 , e− 4 ] ⊂ [−π, π]. Extending
µ as a periodic function of period 2π, it can be expanded into a Fourier series
X
µ(τ ) =
aj eijτ ,
j∈Z
with rapidly decreasing coefficients. Since
X
µ(τ ) =
j∈Z\{0}
P
j
aj = µ(0) = 0, we can write
aj (eijτ − 1) ,
hence, for ξ > 0,
(2.1)
X
m(ξ) =
aj (eije
−ξ
j∈Z\{0}
We set
mj (ξ) = eije
−ξ
−1=
so that
mj (L) =
∞
X
(ij)`
`=1
∞
X
(ij)`
`!
`=1
− 1) .
`!
e−`ξ ,
e−`L .
This makes sense, because the series converges in the operator norm in L(L2 ).
Therefore, the convolution kernel kj of mj (L) equals
(2.2)
kj (z, t) =
∞
X
(ij)`
`=1
`!
p` (z, t) .
Lemma 2.2. We have
kkj k2 ≤ C|j| .
Proof. Let
−ξ
eije − 1
.
m̃j (ξ) =
e−ξ
Observe that m̃j is bounded on R+ , and
km̃j k∞ ≤ |j| .
101
Then, if k̃j is the convolution kernel of m̃j (L) and f ∈ L2 (Hn ),
f ∗ kj = mj (L)f = m̃j (L)e−L f = f ∗ p1 ∗ k̃j .
Therefore,
kj = p1 ∗ k̃j = m̃j (L)p1 .
It follows from Proposition 1.4 in Chapter I that
kkj k2 ≤ km̃j (L)kL(L2 ) kp1 k2 = km̃j k∞ kp1 k2 ≤ C|j| .
Lemma 2.3. There is a constant ν > 0 such that
Z
(2.3)
e (z,t) |kj (z, t)| dz dt ≤ eν|j| .
Hn
Moreover, for every N ∈ N, there is CN > 0 such that
Z
(z, t)N |kj (z, t)| dz dt ≤ CN |j|N +n+2 .
(2.4)
Hn
Proof. By (2.2),
(2.5)
Z
e
Hn
(z,t)
|kj (z, t)| dz dt ≤
Z
∞
X
|j|`
`=1
`!
e
Hn
(z,t)
p` (z, t) dz dt .
By Proposition 1.4,
Z
(z,t) f ∗ g(z, t) dz dt ≤
e
Hn
Z Z
(z,t) f (z, t)(w, u)−1 g(w, u) dw du dz dt
≤
e
Z Hn Z Hn (z 0 ,t0 )(w,u) 0 0 =
e
f (z , t ) g(w, u) dw du dz 0 dt0
H
H
Zn n
Z
(z 0 ,t0 ) 0 0 0 0
(w,u) ≤
e
f (z , t ) dz dt
g(w, u) dw du .
e
Hn
Hn
Therefore, if
ν=
Z
e
Hn
(z,t)
p1 (z, t) dz dt ,
we deduce from the fact that p` = p1 ∗ · · · p1 (` times) that
Z
(z,t)
e
p` (z, t) dz dt ≤ ν ` .
Hn
From (2.5) we obtain the first claimed estimate. We then pass to the second.
Given r > 0, we split
the integral
into the sum of the two integrals, extended to
the set where (z, t) > r and (z, t) < r respectively.
102
CHAPTER V
We use (2.3) to estimate the first integral:
Z
Z
N
(z,t)
N
−ρ
|kj (z, t)| dz dt
e
(z, t) |kj (z, t)| dz dt ≤ sup ρ e
(z,t)>r
(z,t)>r
ρ>r
≤ eν|j| sup ρN e−ρ .
ρ>r
If we impose that r > N , then ρN e−ρ is a decreasing function of ρ, so that
Z
N
(2.6)
(z, t) |kj (z, t)| dz dt ≤ eν|j| r N e−r .
(z,t)>r
To estimate the second integral, we use instead Lemma 2.2 to obtain that
Z
N
(z, t) |kj (z, t)| dz dt ≤
(z,t)<r
Z
12 Z
21
2N
2
≤
(z, t)
dz dt
|kj (z, t)| dz dt
(z,t)<r
(z,t)<r
Z
21
2N
≤ C|j| .
(z, t) dz dt
(z,t)<r
Since
Z
(z,t)<r
∞
X
(z, t)2N dz dt =
(z, t)2N dz dt
−(k+1) r<(z,t)<2−k r
k=0 2
Z
∞
X
−2N k 2N
2
r
dz dt
≤
2−(k+1) r<(z,t)<2−k r
≤
k=0
∞
X
Z
(2.7)
2−2N k r 2N 2−(2n+2)k r 2n+2
k=0
2(N +n+1)
= Cr
we have
Z
,
N
(z, t) |kj (z, t)| dz dt ≤ C|j|r N +n+1 .
(z,t)<r
Putting together (2.6) and (2.7), we have that, for r > N ,
Z
(z, t)N |kj (z, t)| dz dt ≤ eν|j| r N e−r + C|j|r N +n+1 .
Hn
If ν|j| > N , taking r = ν|j|, we have
Z
(z, t)N |kj (z, t)| dz dt ≤ ν N |j|N + Cν N +n+1 |j|N +n+2
Hn
≤ CN |j|N +n+2 .
There are only finitely many remaining values of j, so that (2.4) is proved.
103
Lemma 2.4. Let m and k be as in Theorem 2.1. Then, for every N ∈ N,
Z
N
1 + |(z, t)| |k(z, t)| dz dt ≤ CN kmkC N +n+4 .
Hn
Proof. By (2.1),
k(z, t) =
X
aj kj (z, t) ,
j∈Z\{0}
so that, by (2.4),
Z
Hn
(z, t)N |k(z, t)| dz dt ≤ CN
X
j∈Z\{0}
|aj ||j|N +n+2 .
The conclusion follows from the inequality
|aj | = |µ̂(j)| ≤ |j|−M kµkC M ≤ |j|−M kmkC M ,
valid for every M . The control by the C N +n+4 -norm is not optimal, and it will be improved later.
At this stage we do not need to be more precise than this.
End of the proof of Theorem 2.1. Let m̃(ξ) = eξ m(ξ). Then m̃ satisfies the same
assumptions of m, so that the convolution kernel k̃ of m̃(L) satisfies, by Lemma 2.4,
Z
0
(z, t)N |k̃(z, t)| dz dt ≤ CN km̃kC N +n+4 ≤ CN
kmkC N +n+4 .
Hn
From the identity m(L) = e−L m̃(L) = m̃(L)e−L , it follows that
k = k̃ ∗ p1 = p1 ∗ k̃ .
In particular, k is smooth. To prove that k ∈ S(Hn ) is equivalent to proving
that for every non-commutative polynomial P (Z, Z̄) in the left-invariant vector
fields Zj , Z̄j , P (Z, Z̄)k is rapidly decreasing at infinity. We then have
P (Z, Z̄)k = k̃ ∗ P (Z, Z̄)p1 ,
where P (Z, Z̄)p1 = g is rapidly decreasing at infinity.
(z, t) > 1, we split the convolution
Fix the polynomial P and N ∈ N. For
integral in two parts. If (w, u)| < 12 (z, t), then
so that
(2.8)
Z
(w, u)−1 (z, t) ≥ (z, t) − (w, u) > 1 (z, t) ,
2
k̃ (w, u)g (w, u)−1 (z, t) dw du ≤
(w,u)|< 12 (z,t)
Z
−N
dw du
≤C k̃ (w, u)(w, u)−1 (z, t)
(w,u)|< 12 (z,t)
−N
≤ C 0 (z, t) kk̃k1
−N
.
≤ C 00 kmkC n+4 (z, t)
104
CHAPTER V
Moreover,
Z
k̃ (w, u)g (w, u)−1 (z, t) dw du ≤
(w,u)|> 21 (z,t)
Z
(w, u)N (2.9)
−1
N
≤2 N k̃ (w, u)g (w, u) (z, t) dw du
(w,u)|> 12 (z,t) (z, t)
−N
≤ CkmkC N +n+4 (z, t)
.
Putting together (2.8) and (2.9), we find that
(2.10)
P (Z, Z̄)k(z, t) ≤ CP,N kmkC N +n+4
N ,
1 + (z, t)
and this concludes the proof.
3. Mihlin-Hörmander multipliers of L
We have the tools now to prove the sharp Mihlin-Hörmander theorem for multipliers of L.
We need however to make some preliminary digression on the realization of Hn as
a space of homogeneous type, and on the corresponding Calderón-Zygmund theory.
There are two natural homogeneous-type structures on Hn , a left-invariant and a
right-invariant one, that we denote here as (Hn , m, d` ) and (Hn , m, dr ) respectively.
In both cases the measure m is the Lebesgue measure, and they differ in the choice
of the distance, which in one case is the left-invariant distance
d` (z, t), (w, u) = (w, u)−1 (z, t) ,
and in the other case the right-invariant distance
dr (z, t), (w, u) = (z, t)(w, u)−1 .
The terminology corresponds to the different invariance properties of d` and dr :
while
d` (ζ, τ )(z, t), (ζ, τ )(w, u) = d` (z, t), (w, u) ,
for every (ζ, τ ) ∈ Hn , we have instead
dr (z, t)(ζ, τ ), (w, u)(ζ, τ ) = dr (z, t), (w, u) .
We then have two different notions of Calderón-Zygmund kernel.
Definition. A distribution u ∈ S 0 (Hn ) is a left Calderón-Zygmund kernel on Hn
if
(i) the operator T f = f ∗ u extends to a bounded operator on L 2 (Hn );
(ii) away from the origin, u coincides with a locally integrable function u(z, t),
and there is a constant C > 0 such that, for every (w, u) 6= (0, 0),
Z
(3.1)
u (w, u)(z, t) − u(z, t) dz dt ≤ C .
(z,t)>4(w,u)
105
A distribution u ∈ S 0 (Hn ) is a right Calderón-Zygmund kernel on Hn if
(i) the operator T f = u ∗ f extends to a bounded operator on L 2 (Hn );
(ii) away from the origin, u coincides with a locally integrable function u(z, t),
and there is a constant C > 0 such that, for every (w, u) 6= (0, 0),
Z
(3.2)
u (z, t)(w, u) − u(z, t) dz dt ≤ C .
(z,t)>4(w,u)
The following statement belongs to the general Calderón-Zygmund theory (see
Corollary 3.4 in Chapter II).
Proposition 3.1. If u is a left Calderón-Zygmund kernel on Hn , the operator
T f = f ∗ u is weak-type (1,1) and bounded on Lp (Hn ) for 1 < p ≤ 2.
If u is a right Calderón-Zygmund kernel on Hn , the operator T 0 f = u ∗ f is
weak-type (1,1) and bounded on Lp (Hn ) for 1 2.
In contrast with what we have seen in Rn , the fact that a convolution operator is
bounded on Lp (Hn ) for some p ∈ (1, ∞), does not imply25 that the same operator
0
is also bounded on Lp (Hn ). This is related to the non-commutative structure of
Hn . The correct duality result is as follows.
Proposition 3.2. Let u ∈ S 0 (Hn ) and p ∈ (1, ∞). The operator T f = f ∗ u is
0
bounded on Lp (Hn ) if and only if T 0 g = u ∗ g is bounded on Lp (Hn ). In this case
the two operator norms coincide.
Proof. Take f, g ∈ S(Hn ). If fˇ(z, t) = f (−z, −t), an explicit computation shows
that
Z
hf ∗ u, ǧi =
f ∗ u(z, t)ǧ(z, t) dz dt
Hn
= hu, fˇ ∗ ǧi
ˇ .
= hu ∗ g, fi
Therefore
kT kL(Lp ) =
=
sup
kf kp ≤1,kǧkp0 ≤1
sup
kfˇkp ≤1,kgkp0 ≤1
= kT 0 kL(Lp0 ) .
hf ∗ u, ǧi
ˇ
hu ∗ g, fi
We shall use the following consequence of Propositions 3.1 and 3.2.
25 In
fact this is false in general. This phenomenon is known as “asymmetry” of convolution
operators, and it occurs on many non-commutative groups. It is not known if all infinite locally
compact non-commutative groups exhibit asymmetry. For a large class of l.c. groups, called
amenable and including the Heisenberg group, it is true however that if a convolution operator is
bounded on some Lp , then it is also bounded on L2 .
106
CHAPTER V
Corollary 3.3. Let u be a two-sided Calderón-Zygmund kernel (i.e. it is both left
and right C-Z kernel). Then T f = f ∗ u is bounded on Lp (Hn ) for 1 21 on R+ . Then m
is continuous and bounded, by (6.1) in Chapter II, so that m(L) is bounded on
L2 (Hn ). Hence there exists u ∈ S 0 (Hn ) such that m(L)f = f ∗ u.
We shall prove the following result26 .
Theorem 3.4. Assume that m(ξ) is a Mihlin-Hörmander multiplier on R + of
. If m(L)f = f ∗ u, then u is a two-sided Calderón-Zygmund kernel.
order s > 2n+1
2
Hence m(L) is weak-type (1,1) and bounded on Lp (Hn ) for 1 0. We define
mj (ξ) = m(2−j ξ)η(ξ) ,
so that each mj is supported in 12 , 2 , and
X
(3.4)
m(ξ) =
mj (2j ξ) .
(3.3)
j∈Z
We call uj the distribution on Hn such that mj (L)f = f ∗ uj . Our aim is to
prove that the uj are in fact integrable unctions, and that they satisfy conditions
analogous to those imposed on the ϕj of Theorem 5.3 in Chapter II.
To begin with, we prove that the uj are integrable functions and that also 1 +
ε
|(z, t)| uj (z, t) is integrable for some ε > 0. The following statement is not good
enough, but it is one of the starting points for an interpolation argument, which
will lead to the desired estimates.
Lemma 3.5. Assume that m ∈ H N (R), with N ≥ n+5, and is supported in 14 , 4 .
Then m(L)f = f ∗ u, where u is integrable on Hn and
Z
2(N −n−5)
1 + |(z, t)|
|u(z, t)|2 dz dt ≤ CN kmk2H N .
Hn
Proof. It follows from Lemma 2.4, by a limiting argument, that u is integrable and
Z
N −n−5
1 + |(z, t)|
|u(z, t)| dz dt ≤ CN kmkC N −1 .
Hn
26 This
result has been proved independently by D. Müller and E.M. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. 73 (1994), 413-440, and by W.
Hebisch, Multiplier theorem on generalized Heisenberg groups, Coll. Math. 65 (1993), 231-239.
The two papers contain different extensions to other nilpotent groups.
107
On the other hand, (2.10) implies that
1 + |(z, t)|
N −n−5
|u(z, t)| ≤ CN kmkC N −1 .
By (6.1) in Chapter II, km(j) k∞ ≤ Ckm(j) kH 1 , so that kmkC N −1 ≤ kmkH N .
Putting these inequalities together, we conclude the proof. The other starting point for the interpolation is a sharper estimate for s = 32 .
3
Lemma 3.6. Assume that m ∈ H 2 (R) and is supported in 41 , 4 . Then m(L)f =
f ∗ u, where u is square-integrable on Hn and
Z
4
1 + |(z, t)| |u(z, t)|2 dz dt ≤ Ckmk2 3 .
H2
Hn
Proof. The left-hand side is equivalent to the L2 -norm of u plus the L2 -norm
of w+ u, by (1.6).
We then use the Plancherel formula, recalling that ũ(λ, k) =
m |λ|(2k + n) :
kuk22
=
∼
=
Z
∞ X
n+k−1
k=0
∞
X
n−1
(k + 1)
k=0
∞
X
k=0
n−1
Z
+∞
−∞
+∞
−∞
ũ(λ, k)2 |λ|n dλ
m |λ|(2k + n) 2 |λ|n dλ
(k + 1)n−1
kmk22
n+1
(2k + n)
≤ Ckmk2
3
H2
.
Moreover,
kw+ uk22
=
∼
Z
∞ X
n+k−1
k=0
∞
X
n−1
(k + 1)
n−1
k=0
For λ > 0 we have
Z
+∞
−∞
+∞
−∞
2 n
^
(w
+ u)(λ, k) |λ| dλ
2 n
^
(w
+ u)(λ, k) |λ| dλ .
^
(w
+ u)(λ, k) = ∂λ m λ(2k + n)
k + n
−
m λ((2k + n + 2) − m λ(2k + n)
λ
Z 2
0
= (2k + n)m λ(2k + n) − (k + n)
m0 λ(2k + n + s) ds
0
0
= −n m λ(2k + n)
Z 2
− (k + n)
m0 λ(2k + n + s) − m0 λ(2k + n) ds
0
= A(λ, k) + B(λ, k) .
108
CHAPTER V
We make separate estimates of the contributions of A(λ, k) and B(λ, k) to the
L2 -integral. Clearly,
Z
+∞
0
A(λ, k)2 λn dλ = n2
Z
+∞
0
0
m λ(2k + n) 2 λn dλ
n2
=
(2k + n)n+1
Z
4
1
4
0 2 n
m (λ) λ dλ ,
so that
∞
X
(k + 1)
n−1
k=0
Z
+∞
0
≤C
≤
A(λ, k)2 λn dλ
∞
X
(k + 1)
k=0
Ckm0 k22
≤ Ckmk
3
H2
−2
Z
4
1
4
0 2 n
m (λ) λ dλ
.
The estimate for the contribution of B(λ, k) is more delicate. We have
Z
0
+∞ B(λ, k)2 λn dλ =
Z 2
2 n
0
0
= (k + n)
m λ(2k + n + s) − m λ(2k + n) ds λ dλ
0
0
Z 2 Z +∞ 2
0
2
≤ 2(k + n)
m λ(2k + n + s) − m0 λ(2k + n) λn dλ ds
0
0
2
2 Z 2Z 4
2(k + n)
λs 0
0
=
− m (λ) λn dλ ds
m λ +
n+1
(2k + n)
2k + n
0
0
Z 2Z 4 2
C
λs 0
0
≤
m
λ
+
−
m
(λ)
λ dλ ds
(k + 1)n−1 0 0
2k + n
2λ
Z 4 Z 2k+n
0
C
m (λ + h) − m0 (λ)2 dh dλ
≤
(k + 1)n−2 0 0
8
Z 4 Z 2k+n
0
C
m (λ + h) − m0 (λ)2 dh dλ .
≤
(k + 1)n−2 0 0
2
Z
+∞
109
Summing over k and using the the Plancherel formula in R, we obtain that
Z +∞
∞
X
n−1
B(λ, k)2 λn dλ
(k + 1)
0
k=0
≤C
=C
Z
Z
Z
∞
4X
(k + 1)
0 k=0
0
4Z
4
Z
8
n
0
X
4
k< h
8
n
Z
8
2k+n
0
0
m (λ + h) − m0 (λ)2 dh dλ
2
(k + 1) m0 (λ + h) − m0 (λ) dh dλ
0
m (λ + h) − m0 (λ)2 dh dλ
h2
0
0
Z +∞ Z
0
m (λ + h) − m0 (λ)2 dλ dh
≤C
h2
0
R
Z +∞ Z
dh
=C
τ 2 |m̂(τ )|2 |eihτ − 1|2 dτ 2
h
0
R
Z +∞ ihτ
Z
2
|e − 1|
=C
τ 2 |m̂(τ )|2
dh dτ
h2
0
R
Z +∞ ih
Z
|e − 1|2
3
2
=C
|τ | |m̂(τ )|
dh dτ
h2
0
R
= Ckmk2 3 .
≤C
H2
We have so proved that
∞
X
(k + 1)
k=0
n−1
Z
+∞
0
2 n
2
^
(w
+ u)(λ, k) |λ| dλ ≤ Ckmk
3
H2
.
Similar computations allow to obtain the same estimate for the integral over
negative values of λ. , and is supported in
Corollary
3.7. Assume that m ∈ H s (R), with s > 2n+1
2
1 2n+1
, 2 . If m(L)f = f ∗ u, then, for 0 < ε < s − 2 ,
2
Z
ε
1 + |(z, t)| |u(z, t)| dz dt ≤ Cε kmkH s .
Hn
Proof. Let ψ(ξ) be a smooth function supported in 14 , 4 and identically equal to
1 on 21 , 2 . Given a bounded function m(ξ) on R, let u be the convolution kernel
of (mψ)(L), and let S
be the linear operator given by Sm = u. If, in particular, m
1
is supported in 2 , 2 , then mψ = m, so that Sm is the kernel u in the statement.
α
Denote by L2α (Hn ) the space of functions f on Hn such that 1+|(z, t)| f (z, t) ∈
L2 (Hn ). Then Lemma 3.6 states that
3
S : H 2 (R) −→ L22 (Hn ) ,
and Lemma 3.5 states that if N ≥ n + 5,
S : H N (R) −→ L2N −n−5 (Hn ) .
110
CHAPTER V
A simple modification to the proof of the interpolation Lemma 7.2 in Chapter II
shows that if
3
θ + (1 − θ)N ,
2
s=
α = 2θ + (1 − θ)(N − n − 5) ,
with 0 < θ < 1, then
For fixed s >
S : H s (R) −→ L2α (Hn ) ,
3
2
N −s
N − 32
and N > s, we find θ =
α = αN = 2
, hence
3 N −n−5
N −s
.
3 + (s − )
2 N − 32
N−2
As N → +∞, αN tends monotonically to s + 12 . Therefore
Z
2(s+ 21 −δ)
(3.5)
1 + |(z, t)|
|u(z, t)|2 dz dt ≤ Cδ kmk2H s ,
Hn
for every δ > 0.
Assume now that s >
Z
Hn
ε
2n+1
2
and 0 < ε < s −
1 + |(z, t)| |u(z, t)| dz dt ≤
Z
Hn
×
2n+1
.
2
1 + |(z, t)|
Z
Hn
Then, if δ < s −
2(s+ 12 −δ)
1 + |(z, t)|
2n+1
2
2
− ε,
|u(z, t)| dz dt
−2(s+ 21 −δ−ε)
dz dt
21
21
.
The last integral is convergent, because the exponent is strictly smaller than the
negative of the homogeneous dimension Q = 2n + 2 of Hn . The conclusion follows
from (3.5). We look now for a substitute of condition (c) of Theorem 5.3 in Chapter II on
the uj . We want to replace the ordinary L1 -Lipschitz condition, which concern the
“abelian” differences uj (z + w, t + u) − uj (z, t), with
similar conditions, involving
“non-abelian” differences of the form uj (z, t)(w, u) −uj (z, t) and uj (w, u)(z, t) −
uj (z, t).
1 Lemma 3.8. Assume that m ∈ H s (R), with s > 2n+1
,
and
is
supported
in
2
2, 2 .
If m(L)f = f ∗ u, then
Z
u (z, t)(w, u) − u(z, t) dz dt ≤ CkmkH s (w, u)| ,
(3.6)
Hn
and
(3.7)
Z
Hn
u (w, u)(z, t) − u(z, t) dz dt ≤ CkmkH s (w, u)| .
Proof. The generic increment (w, u) can be written as
p
p
p
p
(w, u) = (w, 0)( |u|e1 , 0)(±i |u|e1 , 0)(− |u|e1 , 0)(∓i |u|e1 , 0) ,
111
where the ± sign depends on the signum of u. This implies that it is sufficient to
prove (3.6) and (3.7) for “horizontal” increments (i.e. with a zero t-component).
In fact, restricting our attention to (3.6) and assuming u > 0 for simplicity, we can
then write
Z
u (z, t)(w, u) − u(z, t) dz dt ≤
Hn
Z
√
√
√
u (z, t)(w, u) − u (z, t)(w, 0)( ue1 , 0)(i ue1 , 0)(− ue1 , 0) dz dt
≤
Hn
Z
u (z, t)(w, 0) − u(z, t) dz dt
+ ···+
Hn
Z
√
u (z, t)(−i ue1 , 0) − u(z, t) dz dt
=
Hn
Z
u (z, t)(w, 0) − u(z, t) dz dt
+ ···+
Hn
√ ≤ CkmkH s |w| + 4 u
≤ C 0 kmkH s (w, u)| .
Suppose therefore that u = 0 in (3.6). By composing if necessary, u with a
unitary transformation of Cn , we can assume that w = re1 , with r > 0. We claim
that we can transform the difference into an integral by the fundamental theorem
of calculus.
ξ
s
Consider
is sup 1 in fact the multiplier m̃(ξ) = e m(ξ) is also in H (Hn ) and
ported in 2 , 2 . If ũ is the corresponding convolution kernel, then ũ ∈ L1 (Hn ), by
Corollary 3.7, and u = ũ ∗ p1 = p1 ∗ ũ. Hence u is C ∞ . Therefore
Z r
d
u (z, t)(re1 , 0) − u(z, t) =
u (z, t)(se1 , 0) ds
ds
(3.8)
Z0 r
=
X1 u (z, t)(se1 , 0) ds .
0
Now, X1 u = ũ ∗ X1 p1 ∈ L1 (Hn ), and
kX1 uk1 ≤ Ckũk1 ≤ C 0 km̃kH s ≤ C 00 kmkH s .
Hence
Z Z r
Z
u (z, t)(re1 , 0) − u(z, t) dz dt =
X1 u (z, t)(se1 , 0) ds dz dt
0
Hn
Hn
Z rZ
X1 u (z, t)(se1 , 0) dz dt ds
≤
0
Hn
= rkX1 uk1 .
In (3.7) the increments are on the left, hence (3.8) must be replaced by
Z r
d
u (se1 , 0)(z, t) ds
u (re1 , 0)(z, t) − u(z, t) =
ds
Z0 r
(r)
=
X1 u (z, t)(se1 , 0) ds ,
0
112
CHAPTER V
(r)
where the right-invariant vector field X1
(r)
appears. The identity
(r)
(r)
X1 u = X1 (p1 ∗ ũ) = (X1 p1 ) ∗ ũ
then leads us to the conclusion.
We consider next condition (b) of Theorem 5.3 in Chapter II.
s
Lemma 3.9. Assume that
R m ∈ H (R), with s >
If m(L)f = f ∗ u, then Hn u(z, t) dz dt = 0.
2n+1
,
2
and is supported in
1
2
,2 .
Proof. Let kj be the kernel in (2.2). By (1.1),
Z
Z
∞
X
(ij)`
kj (z, t) dz dt =
Hn
`=1
`!
p` (z, t) dz dt =
Hn
∞
X
(ij)`
`=1
`!
= eij − 1 .
Since s > 32 , m is C 1 , and so is µ. It follows that the coefficients aj in (2.1) are
summable, because, by the Parseval formula,
X
j∈Z\{0}
Since u =
Z
P
j∈Z\{0}
|aj | ≤
X
j∈Z\{0}
|jaj |
2
12
= kµ0 k2 .
aj kj , we have
X
u(z, t) dz dt =
Hn
j∈Z\{0}
aj (eij − 1) = µ(1) = m(0) = 0 .
Finally, some further remarks are needed, concerning the decomposition (3.4)
of m.
Let mj , with j ∈ Z, be the multiplier in (3.3), with uj such that mj (L)f = f ∗uj .
By Proposition 1.4 in Chapter I, if f, g ∈ L2 (Hn ), then
Z ∞
hm(L)f |gi =
m(λ) dνf,g (λ)
0
XZ ∞
=
mj (2j λ) dνf,g (λ)
(3.9)
j∈Z
=
X
j∈Z
Hence m(L) =
P
j∈Z
0
hmj (2j L)f |gi .
mj (2j L) in the weak topology.
Lemma 3.10. Let
j
(j)
uj (z, t) = 2−(n+1)j uj (2− 2 z, 2−j t) .
(j)
Then mj (2j L)f = f ∗ uj , and
u=
X
j∈Z
(j)
uj
,
113
in the sense of distributions.
Proof. By Theorem 5.6 and (5.10) in Chapter IV,
Z
uej (λ, k) =
uj (z, t)eiλt ψk |λ||z|2 dz dt = mj |λ|(2k + n) .
Hn
Therefore,
g
(j)
uj (λ, k) =
Z
Hn
j
uj (z, t)eiλ2 t ψk |λ|2j |z|2 dz dt = mj 2j |λ|(2k + n) .
(j)
Applying Theorem 5.6 in Chapter IV again, we conclude that uj is the convolution kernel of mj (2j L). By (3.9), if f, g ∈ L2 (Hn ) and g ∗ (z, t) = g(−z, −t),
hu|f ∗ gi = hu ∗ g ∗ |f i
= hm(L)g ∗ |f i
X
=
hmj (2j L)g ∗ |f i
j∈Z
=
X
j∈Z
(j)
huj |f ∗ gi .
Take now ϕ ∈ S(Hn ). We want to prove that
hu|ϕi =
X
j∈Z
(j)
huj |ϕi .
(j)
Since u is radial, hu|ϕi = hu|P ϕi (and the same holds for uj ), where P is the
orthogonal projection from L2 (Hn ) onto L2rad (Hn ) (see (5.7) in Chapter IV). By
Lemma 5.5 in Chapter IV, P ϕ ∈ Srad (Hn ).
It is then sufficient to prove that every ϕ ∈ Srad (Hn ) can be written as ϕ = f ∗ g,
with f, g ∈ L2 (Hn ).
By Proposition 2.7 in Chapter IV and its extension to Hn ,
ϕ̃(λ, k) ≤
CN
1 + |λ|(k + 1)
N ,
for every N ∈ N. Therefore, taking N > n + 1,
Z X
XZ
n
k + n − 1 (k + 1)n−1
n
ϕ̃(λ, k) |λ| dλ ≤ CN
N |λ| dλ
n
−
1
R k∈N
k∈N R 1 + |λ|(k + 1)
< +∞ .
By the Plancherel formula, if we impose that
1
f˜(λ, k) = ϕ̃(λ, k) 2 ,
then f, g ∈ L2 (Hn ) and f ∗ g = ϕ.
1
g̃(λ, k) = ϕ̃(λ, k) 2 arg ϕ̃(λ, k) ,
114
CHAPTER V
We have now all the ingredients for the proof of Theorem 3.4.
P
(j)
Proof of Theorem 3.4. By Lemma 3.10, u = j∈Z uj . By Lemmas 3.7, 3.8, 3.9,
the uj satisfy the following conditions with the same constant C:
ε
R
(i) For some ε > 0, Hn 1 + |(z, t)| |uj (z, t)| dz dt ≤ Cε ;
R
(ii) Hn uj (z, t) dz dt = 0;
(iii) for every (w, u) ∈ Hn ,
Z
Z Hn
Hn
uj (z, t)(w, u) − uj (z, t) dz dt ≤ C (w, u)| ,
uj (w, u)(z, t) − uj (z, t) dz dt ≤ C (w, u)| .
A straightforward adaptation of the proof of Theorem 5.3 in Chapter II shows
that u is a two-sided Calderón-Zygmund kernel. Corollary 3.3 can then be applied.

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