# Anisotropic Hardy Spaces and Wavelets

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*Marcin Bownik*

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 associated with very general discrete groups of dilations. This
formulation includes the classical isotropic Hardy space theory of Fefferman
and Stein and parabolic Hardy space theory of Calderón and
Torchinsky.

Given a dilation \(A\), that is an \(n\times
n\) matrix all of whose eigenvalues \(\lambda\) satisfy
\(|\lambda|>1\), define the radial maximal function
\[M^0_\varphi
f(x): = \sup_{k\in\mathbb{Z}} |(f*\varphi_k)(x)|, \qquad\text{where }
\varphi_k(x) = |\det A|^{-k} \varphi(A^{-k}x).\]
Here \(\varphi\) is
any test function in the Schwartz class with \(\int \varphi \not =0\).
For \(0<p<\infty\) we introduce the corresponding anisotropic Hardy
space \(H^p_A\) as a space of tempered distributions \(f\) such
that \(M^0_\varphi f\) belongs to \(L^p(\mathbb R^n)\).

Anisotropic Hardy spaces 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 \(\varphi\) as long as
\(\int \varphi \not =0\). These spaces can be equivalently introduced in
terms of grand, tangential, or nontangential maximal functions. We prove the
Calderón-Zygmund decomposition which enables us to show the atomic
decomposition of \(H^p_A\). As a consequence of atomic decomposition we
obtain the description of the dual to \(H^p_A\) in terms of Campanato
spaces. We provide a description of the natural class of operators acting on
\(H^p_A\), i.e., Calderón-Zygmund singular integral operators. We
also give a full classification of dilations generating the same space
\(H^p_A\) in terms of spectral properties of \(A\).

In the second part of this paper we show that for every
dilation \(A\) preserving some lattice and satisfying a particular
expansiveness property there is a multiwavelet in the Schwartz class. We also
show that for a large class of dilations (lacking this property) all
multiwavelets must be combined minimally supported in frequency, and thus far
from being regular. We show that \(r\)-regular (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.

#### Table of Contents

# Table of Contents

## Anisotropic Hardy Spaces and Wavelets

- Contents v6 free
- Chapter 1. Anisotropic Hardy Spaces 18 free
- 1. Introduction 18
- 2. The space of homogeneous type associated with the discrete group of dilations 512
- 3. The grand maximal definition of anisotropic Hardy spaces 1118
- 4. The atomic definition of anisotropic Hardy spaces 1926
- 5. The Calderón-Zygmund decomposition for the grand maximal function 2330
- 6. The atomic decomposition of H[sup(p)] 3542
- 7. Other maximal definitions 4249
- 8. Duals of H[sup(p)] 5057
- 9. Calderón-Zygmund singular integrals on H[sup(p)] 6067
- 10. Classification of dilations 7077

- Chapter 2. Wavelets 8289
- Notation Index 116123
- Bibliography 118125