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Anisotropic Hardy spaces and wavelets. (English) Zbl 1036.42020
In this paper, motivated in part by the role of discrete groups of dilations in wavelet theory, the author introduces and investigates the anisotropic Hardy spaces associated with very general discrete groups of dilations. This formulation includes the classic isotropic Hardy space theory of {\it C. Fefferman} and {\it E. M. Stein} [Acta Math. 129, 137--193 (1972; Zbl 0257.46078)] and parabolic Hardy space theory of {\it A. P. Calderón} and {\it A. Torchinsky} [Adv. Math. 16, 1--64 (1975; Zbl 0315.46037) ibid. 24, 101--171 (1977; Zbl 0355.46021)]. Given a {\it dilation} $A$, that is, an $n \times n$ matrix all of whose eigenvalues $\lambda$ satisfy $\vert \lambda\vert >1$, define the radial maximal function $$M^0_{\psi}f(x):= \sup \vert f \ast \psi_k\vert (x), \quad \text{ where} \quad \psi_k(x) := \mid \det A\vert ^{-k} \psi(A^{-k}x).$$ Here $\psi$ is any function in the Schwartz class with $\int \psi \ne 0$. For $0 < p < \infty$, the author introduces the corresponding anisotropic Hardy space $H^p_A$ as a space of tempered distributions $f$ such that $M^0_{\psi}f$ belongs to $L^p(R^n)$. 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 $\psi$ as long as $\int \psi \ne 0$. These spaces can be equivalently introduced in terms of grand, tangential or nontangential maximal functions. The author proves the Calderón--Zygmund decomposition, which enables him to show the atomic decomposition of $H^p_A$. As a consequence of atomic decomposition he obtains the description of the dual to $H^p_A$ in terms of Campanato spaces. He provides a description of the natural class of operators acting on $H^p_A$, i.e., Calderón--Zygmund singular integral operators. He also gives a full classification of dilations generating the same space $H^p_A$ in terms of spectral properties of $A$. In the second part of the paper, the author shows that for every dilation $A$ preserving some lattice and satisfying a particular expansiveness property there is a multiwavelet in the Schwartz class. He also shows that for a large class of dilations (lacking this property) all multiwavelets must be combined minimally supported in frequency, and thus are far from being regular. He shows that $r$--regular (tight frame) multiwavelets form an unconditional basis (tight frame) for the anisotropic Hardy space $H^p_A$. He also describes the sequence space characterizing wavelet coefficients of elements of the anisotropic Hardy space.

##### MSC:
 42B30 $H^p$-spaces (Fourier analysis) 42C40 Wavelets and other special systems 42B20 Singular and oscillatory integrals, several variables 42B25 Maximal functions, Littlewood-Paley theory 46F05 Topological linear spaces of test functions, distributions and ultradistributions