Calderón–Zygmund operators on multiparameter Lipschitz spaces of homogeneous type

Shaoyong He; Jiecheng Chen

doi:10.1515/forum-2021-0204

Article

Calderón–Zygmund operators on multiparameter Lipschitz spaces of homogeneous type

Shaoyong He and Jiecheng Chen

Published/Copyright: December 1, 2021

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From the journal Forum Mathematicum Volume 34 Issue 1

Abstract

The purpose of this paper is to establish a necessary and sufficient condition for the boundedness of general product singular integral operators introduced by Han, Li and Lin [Y. Han, J. Li and C.-C. Lin, Criterion of the L2 boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 2016, 3, 845–907] on the multiparameter Lipschitz spaces of homogeneous type M~=M1×⋯×Mn. Each factor space Mi, 1≤i≤n, is a space of homogeneous type in the sense of Coifman and Weiss. These operators generalize those studied by Journé on the Euclidean space and include operators studied by Nagel and Stein on Carnot–Carathéodory spaces. The main tool used here is the discrete Littlewood–Paley–Stein theory and almost orthogonality together with a density argument for the product Lipschitz spaces in the weak sense.

Keywords: Multiparameter Lipschitz space; discrete Littlewood–Paley–Stein theory; product singular integral operator; space of homogeneous type

MSC 2010: 42B20; 42B25; 46E35

Communicated by Christopher D. Sogge

Funding source: Natural Science Foundation of Zhejiang Province

Award Identifier / Grant number: LQ22A010018

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12071437

Funding statement: This research was funded by Natural Science Foundation of Zhejiang Province (Grant number LQ22A010018) and National Natural Science Foundation of China (Grant number 12071437).

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Received: 2021-08-08

Published Online: 2021-12-01

Published in Print: 2022-01-01

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https://doi.org/10.1515/forum-2021-0204

Keywords for this article

Multiparameter Lipschitz space; discrete Littlewood–Paley–Stein theory; product singular integral operator; space of homogeneous type