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What is Anisotropic Hardy Spaces and Wavelets about?

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 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 n

Who reads Anisotropic Hardy Spaces and Wavelets?

Author

Marcin Bownik

Publisher

Providence, R.I. : American Mathematical Society, c2003.

Published

2003

Language

EN

ISBN

9781470403799

Category

nonfiction

Subjects

Mathematics, Science, Stem