Anisotropic Hardy Spaces of Musielak-Orlicz Type with Applications to Boundedness of Sublinear Operators

Baode Li,Wen Yuan,Dachun Yang

doi:10.1155/2014/306214

Baode Li, Wen Yuan + Show 1 more

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https://doi.org/10.1155/2014/306214

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Abstract

Let φ : ℝn × [0, ∞)→[0, ∞) be a Musielak-Orlicz function and A an expansive dilation. In this paper, the authors introduce the anisotropic Hardy space of Musielak-Orlicz type, H A φ(ℝn), via the grand maximal function. The authors then obtain some real-variable characterizations of H A φ(ℝn) in terms of the radial, the nontangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy space H A p(ℝn) with p ∈ (0,1] and are new even for its weighted variant. Finally, the authors characterize these spaces by anisotropic atomic decompositions. The authors also obtain the finite atomic decomposition characterization of H A φ(ℝn), and, as an application, the authors prove that, for a given admissible triplet (φ, q, s), if T is a sublinear operator and maps all (φ, q, s)-atoms with q < ∞ (or all continuous (φ, q, s)-atoms with q = ∞) into uniformly bounded elements of some quasi-Banach spaces ℬ, then T uniquely extends to a bounded sublinear operator from H A φ(ℝn) to ℬ. These results are new even for anisotropic Orlicz-Hardy spaces on ℝn.

Highlights

The theory of Hardy spaces on the Euclidean space Rn plays an important role in various fields of analysis and partial differential equations
When p ∈ (0, 1], it is well known that Riesz transforms are not bounded on Lp(Rn); they are bounded on Hardy spaces Hp(Rn)
There were several efforts to extend classical Hardy spaces, some of which are weighted anisotropic Hardy spaces [6] associated with general expansive dilations and A∞ Muckenhoupt weights

Summary

Introduction

The theory of Hardy spaces on the Euclidean space Rn plays an important role in various fields of analysis and partial differential equations (see, e.g., [1,2,3,4,5]). We introduce anisotropic Hardy spaces of Musielak-Orlicz type, HAφ(Rn), via grand maximal functions and characterize these spaces via anisotropic atomic decompositions. These Hardy spaces include classical Hardy spaces Hp(Rn) of Fefferman and Stein [1], weighted anisotropic Hardy spaces of Bownik [6], and. The second goal is to obtain some new real-variable characterizations of HAφ(Rn) in terms of the radial, the nontangential, and the tangential maximal functions via some bounded estimates of the truncated maximal function pointwise or in anisotropic Musielak-Orlicz spaces which are motivated by [9, Section 7]. We introduce the anisotropic Hardy spaces of Musielak-Orlicz type, HAφ(Rn), via grand maximal functions, and some basic properties of these spaces are presented. For any a ∈ R, ⌊a⌋ denotes the maximal integer not larger than a

Anisotropic Hardy Spaces of Musielak-Orlicz Type

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