On the kernel conditions of operators mapping atoms to molecules in local Hardy spaces

2025/03/06 by Chun Ho Lau, Lau, Chun Ho, Cláudio Roberto Fóffano Vasconcelos +1
Mathematics · #42B20 #42B30 #42B35 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2503.04604

openalex publication_date 2025/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we explore the relationship between the operators mapping atoms to molecules in local Hardy spaces h^p(ℝⁿ) and the size conditions of its kernel. In particular, we show that if the kernel of a Calderón--Zygmund-type operator satisfies an integral-type size condition and a T^*-type cancellation, then the operator maps h^p(ℝⁿ) atoms to molecules. On the other hand, assuming that T is an integral type operator bounded on L²(ℝⁿ) that maps atoms to molecules in h^p(ℝⁿ), then the kernel of such operator satisfies the same integral-type size conditions. We also provide the L¹(ℝⁿ) to L^1,∞(ℝⁿ) boundedness for such operators connecting our integral-type size conditions on the kernel with others presented in the literature.

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