Littlewood–Paley and Finite Atomic Characterizations of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Jun Liu, Ferenc Weisz, Dachun Yang, Wen Yuan

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

Let p(·):Rn→(0,∞] be a variable exponent function satisfying the globally log-Hölder continuous condition, q∈ (0 , ∞] and A be a general expansive matrix on Rn. Let HAp(·),q(Rn) be the anisotropic variable Hardy–Lorentz space associated with A defined via the radial grand maximal function. In this article, the authors characterize HAp(·),q(Rn) by means of the Littlewood–Paley g-function or the Littlewood–Paley gλ∗-function via first establishing an anisotropic Fefferman–Stein vector-valued inequality on the variable Lorentz space Lp(·),q(Rn). Moreover, the finite atomic characterization of HAp(·),q(Rn) is also obtained. As applications, the authors then establish a criterion on the boundedness of sublinear operators from HAp(·),q(Rn) into a quasi-Banach space. Applying this criterion, the authors show that the maximal operators of the Bochner–Riesz and the Weierstrass means are bounded from HAp(·),q(Rn) to Lp(·),q(Rn) and, as consequences, some almost everywhere and norm convergences of these Bochner–Riesz and Weierstrass means are also obtained. These results on the Bochner–Riesz and the Weierstrass means are new even in the isotropic case.

Original languageEnglish
Pages (from-to)874-922
Number of pages49
JournalJournal of Fourier Analysis and Applications
Volume25
Issue number3
DOIs
Publication statusPublished - Jun 15 2019

Keywords

  • (Hardy-)Lorentz space
  • Bochner–Riesz means
  • Expansive matrix
  • Finite atom
  • Littlewood–Paley function
  • Variable exponent
  • Weierstrass means

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Applied Mathematics

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