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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
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).

References

Adams, D. R. and Xiao, J., ‘Morrey spaces in harmonic analysis’, Ark. Mat. 50 (2012), 201–230.10.1007/s11512-010-0134-0CrossRefGoogle Scholar
Chen, Y., Ding, Y. and Wang, X., ‘Compactness of commutators for singular integrals on Morrey spaces’, Canad. J. Math. 64 (2012), 257–281.10.4153/CJM-2011-043-1CrossRefGoogle Scholar
Di Fazio, G. and Ragusa, M. A., ‘Commutators and Morrey spaces’, Boll. Unione Mat. Ital. 5(7) (1991), 323–332.Google Scholar
Duong, X. T., Lacey, M., Li, J., Wick, B. D. and Wu, Q. Y., ‘Commutators of Cauchy–Szegő type integrals for domains in ${\mathbb{C}}^n$ with minimal smoothness’, Indiana Univ. Math. J. 70(4) (2021), 1505–1541.CrossRefGoogle Scholar
Duong, X. T., Lanzani, L., Li, J. and Wick, B. D., ‘The Cauchy–Szegő projection and its commutator for domains in ${\mathbb{C}}^n$ with minimal smoothness’, Preprint, 2021, arXiv:2005.12740.Google Scholar
Gong, R., Li, J., Pozzi, E. and Vempati, M. N., ‘Commutators on weighted Morrey spaces on spaces of homogeneous type’, Anal. Geom. Metr. Spaces 8(1) (2020), 305–334.10.1515/agms-2020-0116CrossRefGoogle Scholar
Hart, J. and Torres, R. H., ‘John–Nirenberg inequalities and weight invariant BMO spaces’, J. Geom. Anal. 29(2) (2019), 1608–1648.10.1007/s12220-018-0054-yCrossRefGoogle Scholar
Kokilashvili, V. and Meskhi, A., ‘The boundedness of sublinear operators in weighted Morrey spaces defined on spaces of homogeneous type’, in: Function Spaces and Inequalities, Springer Proceedings in Mathematics and Statistics, 206 (eds. P. Jain and H. J. Schmeisser) (Springer, Singapore, 2017), 193–211.10.1007/978-981-10-6119-6_9CrossRefGoogle Scholar
Komori, Y. and Shirai, S., ‘Weighted Morrey spaces and a singular integral operator’, Math. Nachr. 282 (2009), 219–231.10.1002/mana.200610733CrossRefGoogle Scholar
Krantz, S. G. and Li, S.-Y., ‘Boundedness and compactness of integral operators on spaces of homogeneous type and applications, II’, J. Math. Anal. Appl. 258 (2001), 642–657.CrossRefGoogle Scholar
Kronz, M., ‘Some function spaces on spaces of homogeneous type’, Manuscripta Math. 106(2) (2001), 219–248.CrossRefGoogle Scholar
Lanzani, L. and Stein, E., ‘The Cauchy integral in ${\mathbb{C}}^n$ for domains with minimal smoothness’, Adv. Math. 264 (2014), 776–830.10.1016/j.aim.2014.07.016CrossRefGoogle Scholar
Lanzani, L. and Stein, E., ‘The Cauchy–Szegő projection for domains in ${\mathbb{C}}^n$ with minimal smoothness’, Duke Math. J. 166 (2017), 125–176.CrossRefGoogle Scholar
Macías, R. and Segovia, C., A Well Behaved Quasidistance for Spaces of Homogeneous Type, Trabajos de Matemática, 32 (Instituto Argentino de Matemática, Buenos Aires, 1981).Google Scholar
Pradolini, G. and Salinas, O., ‘Commutators of singular integrals on spaces of homogeneous type’, Czechoslovak Math. J. 57 (2007), 75–93.10.1007/s10587-007-0045-9CrossRefGoogle Scholar
Tao, J., Yang, D. and Yang, D., ‘Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces’, Math. Methods Appl. Sci. 42(2) (2019), 1631–1651.10.1002/mma.5462CrossRefGoogle Scholar
Tao, J., Yang, D. and Yang, D., ‘Beurling–Ahlfors commutators on weighted Morrey spaces and applications to Beltrami equations’, Potential Anal. 53 (2020), 1467–1491.CrossRefGoogle Scholar