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 × n matrix all of whose eigenvalues λ satisfy |λ| > 1, define the radial maximal functionwhereHere ϕ is any test function in the Schwartz class with ϕ = 0. For 0 < p < ∞ we introduce the corresponding anisotropic Hardy space. 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 ϕ as long as ϕ = 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.

Using the range function approach to shift invariant spaces in L 2 (R n ) we give a simple characterization of frames and Riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l 2 (Z n ). This in turn gives a simplified approach to the analysis of frames and Riesz families done by Gramians and dual Gramians. We prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator. As an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function. Academic Press

No abstract

Abstract. Weighted anisotropic Triebel-Lizorkin spaces are introduced and studied with the use of discrete wavelet transforms. This study extends the isotropic methods of dyadic ϕ-transforms of Jawerth (1985, 1989) to non-isotropic settings associated with general expansive matrix dilations and A ∞ weights.In close analogy with the isotropic theory, we show that weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the ϕ-transforms in appropriate sequence spaces. We also introduce non-isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces, thus extending isotropic results of Frazier and Jawerth.