Anisotropic Hardy Spaces and Wavelets
Marcin Bownik
Author address:
Department of Mathematics, University of Michigan, 525 East Uni-
versity Ave., Ann Arbor, MI 48109
E-mail address: marbow@umich.edu
1
viii ANISOTROPIC HARDY SPACES AND WAVELETS
Abstract
In this paper, motivated in part by the role of discrete groups of dilations in
wavelet theory, we introduce and investigate the anisotropic Hardy spaces associ-
ated with very general discrete groups of dilations. This formulation includes the
classical isotropic Hardy space theo ry of Fefferman and Stein and pa rabolic Ha rdy
space theory of C alder´on and Tor chinsky.
Given a dilation A, that is an n × n matrix all of whose eigenvalues λ satisfy
|λ| > 1, define the radial maximal function
M
0
ϕ
f(x) := sup
k∈Z
|(f ∗ ϕ
k
)(x)|, where ϕ
k
(x) = |det A|
−k
ϕ(A
−k
x).
Here ϕ is any test function in the Schwartz class with
R
ϕ 6= 0. For 0 < p < ∞ we
introduce the corresponding anisotropic Hardy space H
p
A
as a space of temper e d
distributions f such that M
0
ϕ
f belongs to L
p
(R
n
).
Anisotropic Ha rdy spac e s enjoy the basic properties of the classical Hardy
spaces. For example, it turns out that this definition does not depend on the
choice of the test function ϕ as long a s
R
ϕ 6= 0. These spaces can be equivalently
introduced in terms o f grand, tangential, or nontangential maximal functions. We
prove the Calder´on-Zygmund decomposition which enables us to show the atomic
decomposition of H
p
A
. As a conseq uence of atomic decomposition we obtain the
description of the dual to H
p
A
in terms of Campanato spaces. We provide a de-
scription of the natural class of operators acting on H
p
A
, i.e., Calder´on-Zygmund
singular integral operators. We also give a full classification of dilations generating
the same space H
p
A
in terms of spec tral properties of A.
In the second part of this paper we s how that for every dilation A preserving
some lattice and satisfying a particular expansiveness property there is a multi-
wavelet in the Schwartz class. We also show that for a large class of dilatio ns
(lacking this property) all multiwavelets must be combined minimally supported in
frequency, and thus far from being regular. We show that r-regula r (tight frame)
multiwavelets form an unconditional basis (tight frame) for the anisotropic Hardy
space H
p
A
. We also describe the sequence space characterizing wavelet coefficients
of elements of the anisotropic Hardy space.
2000 Mathematics Subject Classification. Primary 42B30, 42C40; Secondary 42B20, 42B25.
Key words and phrases. anisotropic Hardy space, radial maximal function, nontangential
maximal function, grand maximal function, C al der´on-Zygmund decomposition, atomic decompo-
sition, Calder´on-Zygmund operator, Campanato space, wavelets, frame, unconditional basis.
The author thanks his advisor, Prof. Richard Rochberg, for his support, guidance and
encouragement.
ANISOTROPIC HARDY SPACES AND WAVELETS ix
Contents
Chapter 1. Anisotropic Hardy Spaces
1. Introduction 1
2. The space of homog e neous type associated with the discrete group of
dilations 4
3. The gr and maximal definition of anis otropic Hardy spaces 10
4. The atomic definition of anisotropic Hardy s paces 18
5. The Calder ´on-Zygmund decomposition for the gr and maximal function 22
6. The atomic decomposition of H
p
34
7. Other maximal definitions 41
8. Duals of H
p
48
9. Calder´on-Z ygmund singular integrals on H
p
58
10. Clas sification of dilatio ns 68
Chapter 2. Wavelets
1. Introduction 80
2. Wavelets in the Schwartz class 83
3. Limitations on orthogonal wavelets 87
4. Non-orthogonal wavelets in the Schwartz class 92
5. Regular wavelets as an unconditional basis for H
p
94
6. Characterization of H
p
in terms of wavelet coefficients 102
Notation Index 114
Bibliography 116
1. INTRODUCTION 1
CHAPTER 1
Anisotropic Hardy spaces
1. Introduction
In the first chapter of this monog raph we develop the real variable theory of
Hardy spaces H
p
(0 < p < ∞) on R
n
in what we believe is greater generality than
has ever been done.
Historical Background. The theory of Hardy spaces is very rich with many
highly developed branches. A recent inquiry in MathSciNet
r
(the database of
Mathematical Reviews since 1940) revealed a fast growing collection of more than
2800 papers related to s ome extent to various Hardy spaces. Therefore, we can
sketch only the most s ignificant highlights of this theory.
Initially, Hardy spaces origina ted in the context of complex function theory and
Fourier analysis in the beginning of twentieth century. The classical Har dy space
H
p
, where 0 < p < ∞, consists of holomorphic functions f defined on the unit disc
such that
||f||
H
p
:= sup
0<r<1
Z
1
0
|f(re
2πiθ
)|
p
dθ
1/p
< ∞,
or on the upper half plane such that
||f||
H
p
:= sup
0<y<∞
Z
∞
−∞
|f(x + iy)|
p
dx
1/p
< ∞.
If p = ∞ we replace the above integrals by the s uprema. For a systematic expo sition
of the subject see books by Dure n [Du], Garnett [Ga], and Koosis [Ko].
The possible generalizations of these spaces to higher dimensions include Hardy
spaces on the unit ball in C
n
, on the polydisc or on the tube domains over cones, see
books of Rudin [Ru1, Ru2], and Stein and Weiss [SW2]. Another possibility is to
consider space s of conjugate harmonic functions f = (u
0
, . . . , u
n
) on R
n
× (0, ∞),
satisfying certain natural generalizations of the Cauchy-Riemann equations and the
size condition
||f||
H
p
:= sup
0<y<∞
Z
R
n
|f(x
1
, . . . , x
n
, y)|
p
dx
1
. . . dx
n
1/p
< ∞,
see Stein and Weiss [SW1, SW2]. In this development the attention is fo c used on
the boundary values of the harmonic functions, which ar e distributions on R
n
. The
harmonic functions can be then recovered from the boundary values by Poisson
integral for mula. The resulting spaces H
p
(R
n
) are equivalent to L
p
(R
n
) for p > 1.
However, for p ≤ 1 these spaces differ from L
p
(R
n
) and are better suited for the
purp oses of harmonic analy sis than L
p
(R
n
). Indeed, singular integral operators and
multiplier o perators turn o ut to be bounded on H
p
, se e Stein [St1].
The beginning of the 1970’s marked the birth of real-variable theory of Hardy
spaces as we know it today. First, Burkholder, Gundy, and Silverstein in [B GS]
using Brownian motion methods showed that f belongs to the classical Hardy space
2 1. ANISOTROPIC HARDY SPACES
H
p
if and only if the nontangential maximal function of Re f belongs to L
p
. The
real breakthrough came in the work of C. Fefferman and Stein [FS2]. They showed
that H
p
in n dimensions ca n be defined as the space of tempered distributions f
on R
n
whose radial maximal function M
0
ϕ
or nontangential maximal function M
ϕ
belong to L
p
(R
n
), where
M
0
ϕ
f(x) := sup
0<t<0
|(f ∗ ϕ
t
)(x)|,
M
ϕ
f(x) := sup
0<t<∞
sup
|x−y|<t
|(f ∗ ϕ
t
)(y)|,
and ϕ
t
(x) := t
−n
ϕ(x/t). Here ϕ is any test function in the Schwartz class with
R
ϕ 6= 0 or the Poisson kernel ϕ(x) = (1 + |x|
2
)
−(n+1)/2
(in this case f is restricted
to bounded distribution) and the definition of H
p
does not depend on this choice.
To prove this impre ssive result C. Fefferman and Stein introduced a very important
tool, the grand maximal function, which can also be used to define Hardy spaces H
p
.
The real analysis methods also played a decisive role in the well-known C. Fefferman
duality theorem between H
1
and BMO—the space of functions of bounded mean
oscillation.
Further insight into the theory of Hardy space s came from the works of Coifman
[Co] (n = 1) and Latter [La] (n ≥ 1 ) where the atomic decomposition of elements in
H
p
(R
n
) (p ≤ 1) was e xhibited. Atoms are compactly supported functions satisfying
certain boundedness properties a nd some number o f vanishing moments. The Hardy
space H
p
(R
n
) can be thought of in terms of atoms and many impo rtant theorems
can be reduced to easy verifications of statements for atoms.
Other developments followed. Coifman and Weiss in [CW2] intro duced Hardy
spaces H
p
for the general class of spaces of homogeneous type using as a definition
atomic decompositions. Since there is no natural substitute for polyno mials, the
Hardy spaces H
p
on spaces of homogeneous type can be defined only for p ≤ 1
sufficiently clo se to 1. Another approach started in the work of Calder´on and
Torchinsky [CT1, CT2] who developed theory of Hardy spaces H
p
(0 < p < ∞) on
R
n
for nonisotropic dilations. The theor y of Hardy spaces was also established on
more general groups than R
n
. For the Heisenberg group it was done by Geller [Ge]
and for gener al homogeneous groups by Folla nd and Stein [FoS]. We also mention
the development of Hardy space on subsets of R
n
by Jonsson and Wallin [JW] and
weighted Hardy spaces on R
n
by Str¨omberg and Torchinsky in [ST1, ST2].
Parabolic Hardy spaces. Calder´on and Tor chinsky initiated the study o f
Hardy spaces on R
n
with nonisotropic dilations in [CT1, CT2]. They start with a
one parameter continuous subgro up of GL(R
n
, n) of the form {A
t
: 0 < t < ∞}
satisfying A
t
A
s
= A
ts
and
t
α
|x| ≤ |A
t
x| ≤ t
β
|x| for all x ∈ R
n
, t ≥ 1,
for some 1 ≤ α ≤ β < ∞. The infinitesimal generato r P of A
t
= t
P
:= exp(P ln t)
satisfies hP x, xi ≥ hx, xi, where h·, ·i is the standar d scalar product in R
n
. The
induced nonisotropic norm ρ on R
n
satisfies ρ(A
t
x) = tρ(x). The parabolic Hardy
space H
p
(0 < p < ∞) is defined as a space of tempered distributions f whose
nontangential function M
ϕ
f belongs to L
p
(R
n
), where
M
ϕ
f(x) := sup
ρ(x−y)<t
|(f ∗ ϕ
t
)(x)|, ϕ
t
(x) = t
− tr P
ϕ(A
−1
t
x),
