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.