Anisotropic Hardy spaces and wavelets

The classical Hardy space theory of Fefferman and Stein considers function spaces on the Euclidean space associated with the traditional isotropic dilation structure. The extension to nonisotropic case was done by Calderon and Torchinsky who introduced parabolic Hardy spaces associated with other groups of dilations. Further aspects of the theory of Hardy spaces associated with nonisotropic dilations were developed systematically by Folland and Stein motivated by, among other things, their study of harmonic analysis on nilpotent groups. However, these sets of results together do not cover what seems to be the natural question of developing Hardy space theory with respect to general dilations. In this dissertation, 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 previous ones. Given a dilation A, that is an n x n matrix all of whose eigenvalues lambda satisfy |lambda| > 1, define the radial maximal function M04fx :=sup k∈Z f*4k x, where 4kx =detA -k4A-kx . Here 4 is any test function in Schwartz class with 4 ≠ 0. For 0 < p < infinity we introduce the corresponding anisotropic Hardy space HpA as a space of tempered distributions f such that M04f belongs to LpRn .; 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 4 as long as 4 ≠ 0. These spaces can be equivalently introduced in terms of grand, tangential, or nontangential maximal functions. We prove the Calderon-Zygmund decomposition which enables us to show the atomic decomposition of HpA . As a consequence of atomic decomposition we obtain the description of the dual to HpA in terms of Campanato spaces. We provide a description of the natural class of operators acting on HpA i.e., Calderon-Zygmund singular integral operators. We also give a full classification of dilations generating the same space HpA in terms of spectral properties of A.; In the second part of the dissertation we show that for every dilation A preserving some lattice and natural number r there is an r-regular wavelet basis all of whose moments vanish. Such wavelets form an unconditional basis for the anisotropic Hardy space HpA . We also describe the sequence space characterizing wavelet coefficients of elements of the anisotropic Hardy space.