Molecular Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

Jun Liu,Chenlong Yue,Long Huang

doi:10.3390/math9182216

Jun Liu, Chenlong Yue + Show 1 more

Open Access

https://doi.org/10.3390/math9182216

Copy DOI

Abstract

Let p→∈(0,∞)n be an exponent vector and A be a general expansive matrix on Rn. Let HAp→(Rn) be the anisotropic mixed-norm Hardy spaces associated with A defined via the non-tangential grand maximal function. In this article, using the known atomic characterization of HAp→(Rn), the authors characterize this Hardy space via molecules with the best possible known decay. As an application, the authors establish a criterion on the boundedness of linear operators from HAp→(Rn) to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund operators on HAp→(Rn). In addition, the boundedness of anisotropic Calderón–Zygmund operators from HAp→(Rn) to the mixed-norm Lebesgue space Lp→(Rn) is also presented. The obtained boundedness of these operators positively answers a question mentioned by Cleanthous et al. All of these results are new, even for isotropic mixed-norm Hardy spaces on Rn.

Highlights

IntroductionThis article is devoted to exploring the molecular characterization of the anisotropic
This article is devoted to exploring the molecular characterization of the anisotropic~p mixed-norm Hardy space H A (Rn ) from [1], where ~p ∈ (0, ∞)n is an exponent vector and A is a general expansive matrix on Rn. Let HAp (Rn)
H A (Rn ) to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund oper~p ators on H A (Rn )

Summary

Introduction

This article is devoted to exploring the molecular characterization of the anisotropic. ~p theory of anisotropic mixed-norm Hardy spaces H A (Rn ), in this article, we characterize. Zygmund operators from H A (Rn ) to the mixed-norm Lebesgue space L~p (Rn ) (see Theorem 2 below) or to itself (see Theorem 3 below) For this purpose, by the known finite atomic char~p acterization of H A (Rn ), we first give the proof of Theorem 2. To prove Theorem 3, we obtain a technical lemma, which shows that, if T is an anisotropic Calderón-Zygmund operator of orderas in Definition 11, for any (~p, r, `)-atom e a, T (e a) is a harmless constant multiple of a (~p, q, s0 , ε)-molecule with s0 and ε, respectively, as in Definition 11 and (24). Throughout this article, the symbol C ∞ (Rn ) denotes the set of all infinitely differentiable functions on Rn

Preliminaries

Some Applications

Conclusions