Anisotropic Hardy spaces and wavelets

Abstract

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.

Anisotropic Hardy-Lorentz spaces and their applications

“…where H p 0 A (R n ) denotes the anisotropic Hardy space introduced by Bownik in [9]. From the above estimates, we further deduce that…”

“…In this subsection, by using the Calderón-Zygmund decomposition associated with the non-tangential grand maximal function on anisotropic R n established in [9], we obtain the atomic characterizations of H p,q A (R n ). We begin with the following notion of anisotropic (p, r, s)-atoms from [9, p. 19 (R n ) is defined to be the set of all distributions f ∈ S ′ (R n ) satisfying that there exist a sequence of (p, r, s)-atoms, {a k i } i∈N, k∈Z , supported on {x k i + B k i } i∈N, k∈Z ⊂ B, respectively, and a positive constant C such that i∈N χ x k i +B k i (x) ≤ C for all x ∈ R n and k ∈ Z, and f = k∈Z i∈N λ k i a k i in S ′ (R n ), where λ k i ∼ 2 k |B k i | 1/p for all k ∈ Z and i ∈ N with the implicit equivalent positive constants independent of k and i.…”

Two-Level-Split Decomposition of Anisotropic Besov Spaces

“…Then it is easy to see that the affine transforms A v,m (x) := A −m x + v, v ∈ R n , m ∈ Z, define a semi-continuous ellipsoid cover (dilations) in the sense of Definition 2.5. This particular kind of dilations are used in [1,2,3] for the development of anisotropic Hardy, Besov, and Triebel-Lizorkin spaces.…”

“…Calderón and Torchinsky [4,5] have developed the so called parabolic Hardy spaces induces by certain one parameter dilation groups, which were further extended by Folland and Stein [12]. Bownik [1,2] and Bownik and Ho [3] have considered dilations on R n induced by the powers A j of an expansive matrix A with eigenvalues λ satisfying |λ| > 1, and applied them to the development of anisotropic Besov and Triebel-Lizorkin spaces.…”

“…This approach enables us indeed to develop the anisotropic Besov spaces (B-spaces)Ḃ α pq and B α pq for arbitrarily large α > 0 which is the principle difference between our theory and the general theory of Besov spaces on homogeneous spaces. Our B-spaces can be viewed as a generalization of the classical Besov spaces as well as the anisotropic Besov spaces studied by Bownik [1] with weight 1 (see §5.3). For the theory of classical Besov spaces we refer the reader to [16,18,19].…”