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BOUNDEDNESS AND COMPACTNESS OF CAUCHY-TYPE INTEGRAL COMMUTATOR ON WEIGHTED MORREY SPACES
Published online by Cambridge University Press: 08 March 2022
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 $bD$ .
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- © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
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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