Real-Variable Characterizations of New Anisotropic Mixed-Norm Hardy Spaces

2019/10/10 by Huang, Long, Liu, Jun, Yang, Dachun +1
#42B20 #42B25 #46E30 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 42B35 #Secondary 42B30

paper · doi:10.48550/arxiv.1910.05142

Abstract

Let p∈(0,∞)ⁿ and A be a general expansive matrix on ℝⁿ. In this article, via the non-tangential grand maximal function, the authors first introduce the anisotropic mixed-norm Hardy spaces H_A^p(ℝⁿ) associated with A and then establish their radial or non-tangential maximal function characterizations. Moreover, the authors characterize H_A^p(ℝⁿ), respectively, by means of atoms, finite atoms, Lusin area functions, Littlewood-Paley g-functions or g_λ^∗-functions via first establishing an anisotropic Fefferman-Stein vector-valued inequality on the mixed-norm Lebesgue space L^p(ℝⁿ). In addition, the authors also obtain the duality between H_A^p(ℝⁿ) and the anisotropic mixed-norm Campanato spaces. As applications, the authors establish a criterion on the boundedness of sublinear operators from H_A^p(ℝⁿ) into a quasi-Banach space. Applying this criterion, the authors then obtain the boundedness of anisotropic convolutional δ-type and non-convolutional β-order Calderón-Zygmund operators from H_A^p(ℝⁿ) to itself [or to L^p(ℝⁿ)]. As a corollary, the boundedness of anisotropic convolutional δ-type Calderón-Zygmund operators on the mixed-norm Lebesgue space L^p(ℝⁿ) with p∈(1,∞)ⁿ is also presented.

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