Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calderón--Zygmund Operators

2022/01/23 by Jingsong Sun, Dachun Yang, Sun, Jingsong +3 · 1 citation
Mathematics · #30L99 #42B20 #42B35 #46E36 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Primary 42B30 #Secondary 42B25

paper · pdf · doi:10.48550/arxiv.2201.09264

openalex publication_date 2022/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (\mathbbX,d,μ) be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and X(\mathbbX) a ball quasi-Banach function space on \mathbbX. In this article, the authors introduce the weak Hardy space WH_X(\mathbbX) associated with X(\mathbbX) via the grand maximal function, and characterize WH_X(\mathbbX) by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors use the Aoki--Rolewicz theorem and both the dyadic system and the exponential decay of approximations of the identity on \mathbbX, which closely connect with the geometrical properties of \mathbbX, to overcome the difficulties caused by the absence of both the triangle inequality of ‖⋅‖_X(\mathbbX) and the reverse doubling assumption of the measure μ under consideration, and also use the relation between the convexification of X(\mathbbX) and the weak space WX(\mathbbX) associated with X(\mathbbX) to prove that the infinite summation of atoms converges in the space of distributions on \mathbbX. Moreover, all these results have a wide range of generality and, particularly, even when they are applied to the weighted Lebesgue space, the Orlicz space, and the variable Lebesgue space, the obtained results are also new and, actually, some of them are new even on RD-spaces (namely, spaces of homogeneous type satisfying the additional reverse doubling condition).

Cited by

• Estimates for Schrödinger Groups and Imaginary Power Operators on Weak Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces

Related

• Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces I: Decompositions with Applications to Boundedness of Calderón--Zygmund Operators
• Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Characterizations of Maximal Functions, Decompositions, and Dual Spaces
• Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood--Paley Characterizations and Real Interpolation
• Applications of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces
• Hardy spaces, Regularized BMO spaces and the boundedness of Calderón-Zygmund operators on non-homogeneous spaces