Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces

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.

Directional Thermodynamic Formalism

“…, then Definition 2 and the partial ordering property (15) yield the third point. [60,61] using an anisotropic Littlewood Paley analysis. We will drop ∞.…”

“…In that case, signals present anisotropies quantified through regularity characteristics and features that strongly differ when measured in different directions [17,19,46,47,49,[54][55][56]. Classes of functions that satisfy different scaling properties according the coordinate axes have been studied in [29,42,44,46,[49][50][51][53][54][55][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72].…”

Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators

“…The theory of Hardy spaces associated to expansive dilations was recently developed in [2,5]. The other direction of extending classical function spaces is the study of weighted function spaces associated with general Muckenhoupt weights; see [4,[6][7][8][9]18]. García-Cuerva [18] and Strömberg and Torchinsky [33] established a theory of weighted Hardy spaces on R n .…”

Molecular characterization of anisotropic weak Musielak-Orlicz Hardy spaces and their applications

“…The following lemma is a well-known superposition principle, which comes from [9, Lemma 7.13]. When A := 2I n×n , ϕ(x, t) := t p for all x ∈ R n and t ∈ (0, ∞) with p ∈ (0, 1), it goes back to the well-known superposition principle of weak type estimates obtained by Stein (0, 1), where I(ϕ) is as in (6). Assume that {a j } j∈Z+ is a sequence of measurable functions and {λ j } j∈Z+ ⊂ C such that there exists a sequence {x j + B lj } j∈Z+ of dilated balls, where l j ∈ Z, it holds that…”

“…is an anisotropic growth function if w is a classical or an anisotropic A ∞ (A) Muckenhoupt weight (see, for example, [6]) and Φ of lower type p for some p ∈ (0, 1] and of upper type 1. More examples of growth functions can be found in [26,28,27,33].…”