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BOUNDEDNESS AND COMPACTNESS OF CAUCHY-TYPE INTEGRAL COMMUTATOR ON WEIGHTED MORREY SPACES

Part of: Harmonic analysis in several variables Holomorphic functions of several complex variables

Published online by Cambridge University Press:  08 March 2022

RUMING GONG ,
MANASA N. VEMPATI ,
QINGYAN WU  and
PEIZHU XIE
RUMING GONG
Affiliation:
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, PR China e-mail: gongruming@gzhu.edu.cn
MANASA N. VEMPATI
Affiliation:
Department of Mathematics, Washington University–St. Louis, St. Louis, MO 63130-4899, USA e-mail: m.vempati@wustl.edu
QINGYAN WU*
Affiliation:
Department of Mathematics, Linyi University, Shandong 276005, PR China
PEIZHU XIE
Affiliation:
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, PR China e-mail: xiepeizhu@gzhu.edu.cn
*
e-mail: wuqingyan@lyu.edu.cn
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Abstract

In this paper we study boundedness and compactness characterizations of the commutators of Cauchy type integrals on bounded strongly pseudoconvex domains D in $\mathbb C^{n}$ with boundaries $bD$ satisfying the minimum regularity condition $C^{2}$ , based on the recent results of Lanzani–Stein and Duong et al. We point out that in this setting the Cauchy type integral is the sum of the essential part which is a Calderón–Zygmund operator and a remainder which is no longer a Calderón–Zygmund operator. We show that the commutator is bounded on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1<p<\infty $ ) if and only if b is in the BMO space on $bD$ . Moreover, the commutator is compact on the weighted Morrey space $L_{v}^{p,\kappa }(bD)$ ( $v\in A_{p}, 1<p<\infty $ ) if and only if b is in the VMO space on $bD$ .

Keywords

commutator of Cauchy type integralstrongly pseudoconvex domainBMO spaceVMO spaceweighted Morrey space

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B35: Function spaces arising in harmonic analysis 32A50: Harmonic analysis of several complex variables 32A55: Singular integrals

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 113 , Issue 1 , August 2022 , pp. 36 - 56
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Ji Li

The first author was supported by the State Scholarship Fund of China (No. 201908440061). The third author (corresponding author) was supported by NSFC (Nos. 12171221 and 12071197) and NSFS (Nos. ZR2021MA031, ZR2019YQ04 and 2020KJI002).

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