Abstract
Let \(\Omega \) be a domain which belongs to a class of bounded weakly pseudoconvex domains of finite type in \({\mathbb {C}}^n\), let \(d\lambda \) be the Monge–Ampère boundary measure on \(b\Omega \) and \(\varrho \ge 0\) be a non-decreasing function. The aim of this paper is to establish the characterizations of boundedness and compactness for the commutator operators of Cauchy–Fantappiè type integrals with \(L^1(b\Omega ,d\lambda )\) functions on the generalized Morrey spaces \(L^{p}_\varrho (b\Omega ,d\lambda )\), with \(p\in (1, \infty )\).
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Arai, H., Mizuhara, T.: Morrey spaces on spaces of homogeneous type and estimates for \(\Box _b\) and the Cauchy–Szegö projection. Math. Nachr. 186, 5–20 (1997)
Beatrous, F., Li, S.-Y.: Boundedness and compactness of operators of Hankel type. J. Funct. Anal. 111, 350–379 (1993)
Bellaïche, A., Risler, J.J.: Sub-Riemannian Geometry, Progress in Mathematics, vol. 144. Birkhauser, Boston (1996)
Bonami, A., Lohoué, N.: Projecteurs de Bergman et Szegö pour une classe de domaines faiblement pseudo-convexes et estimations \(L^p\). Compos. Math. 46(2), 159–226 (1982)
Bramanti, M., Cerutti, M.C.: Commutators of singular integrals on homogeneous spaces. Boll. Un. Mat. Ital. B (7) 10, 843–883 (1996)
Calderón, A.P., Zygmund, A.: Singular integral operators and differential equations. Am. J. Math. 79, 901–921 (1957)
Campanato, S.: Proprietà di hölderianità di alcune classi di funzioni. Ann. Scuola Norm. Sup. Pisa (3) 17, 175–188 (1963)
Chen, Y., Ding, Y., Wang, X.: Compactness of commutators for singular integrals on Morrey spaces. Can. J. Math. 64(2), 257–281 (2012)
Coifman, R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)
Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103(3), 611–635 (1976)
Duong, X.T., Xiao, J., Yan, L.: Old and new Morrey spaces with heat Kernel bounds. J. Fourier Anal. Appl. 13(1), 87–111 (2007)
Duong, X.T., Lacey, M., Li, J., Wick, B.D., Wu, Q.: Commutators of Cauchy type integrals for domains in \({\mathbb{C}}^n\) with minimal smoothness, to appear on Indiana Univ. Maths. J. (2019)
Fazio, G.D., Ragusa, M.A.: Commutators and Morrey spaces. Boll. Un. Mat. Ital. A (7) 5(3), 323–332 (1991)
Fefferman, C.: The Bergman kernel and biholomorphic mapping of pseudoconvex domains. Invent. Math. 26, 1–65 (1974)
Hansson, T.: On Hardy spaces in complex ellipsoids. Ann. Inst. Fourier (Grenoble) 49(5), 1477–1501 (1999)
Janson, S.: Mean oscillation and commutators of singular integral operators. Ark. Mat. 16, 263–270 (1978)
Krantz, S.G.: Function theory of several complex variables. Reprint of the 1992 edition. AMS Chelsea Publishing, Providence (2001). xvi+564 pp. ISBN: 0-8218-2724-3
Krantz, S.G.: Harmonic and Complex Analysis in Several Variables. Springer Monographs in Mathematics. Springer, Cham (2017). xii+424 pp. ISBN: 978-3-319-63229-2
Krantz, S.G., Li, S.-Y.: Boundedness and compactness of integral operators on spaces of homogeneous type and applications, I. J. Math. Anal. Appl. 258, 629–641 (2001)
Krantz, S.G., Li, S.-Y.: Boundedness and compactness of integral operators on spaces of homogeneous type and applications, II. J. Math. Anal. Appl. 258, 642–657 (2001)
Leray, J.: Le calcul différentiel et intégral sur une variété analytique complexe. Bull. Soc. Math. France 87, 81–180 (1959)
Lerner, A.K., Ombrosi, S., Rivera-Riós, I.P.: Commutators of singular integrals revisited. Bull. Lond. Math. Soc. 51(1), 107–119 (2019)
Li, J., Trang, N.T.T., Leslay, A.W., Brett, D.W.: The Cauchy integral, bounded and compact commutators. Studia Math. 250, 193–216 (2020)
McNeal, J.D.: Estimates on the Bergman kernel of convex domains. Adv. Math. 109, 108–139 (1994)
McNeal, J.D.: The Bergman projection as a singular integral operator. J. Geom. Anal. 4, 91–103 (1994)
McNeal, J.D., Stein, E.M.: The Szegö projection on convex domains. Math. Z. 224, 519–553 (1997)
Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43(1), 126–166 (1938)
Morrey, C.B.: Some recent developments in the theory of partial differential equations. Bull. Am. Math. Soc. 68, 279–297 (1962)
Morrey, C.B.: Partial regularity results for non-linear elliptic systems. J. Math. Mech. 17, 649–670 (1967/1968)
Nagel, A.: Analysis and geometry on Carnot–Carathéodory spaces (2005). http://www.math.wisc.edu/~nagel/2005Book.pdf
Nagel, A., Rosay, J.P., Stein, E.M., Wainger, S.: Estimates for the Bergman and Szegö kernels in \({\mathbb{C}}^2\). Ann. Math. 129, 113–149 (1989)
Pérez, C.: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128, 163–185 (1995)
Stein, E.M.: Three projection operators in complex analysis. In: Colloquium De Giorgi 2010–2012, pp. 49–59
Tao, J., Yang, D., Yang, D.: Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces. Math. Methods Appl. Sci. 42, 1631–1651 (2019)
Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Dover Publications, New York (2004)
Uchiyama, A.: On the compactness of operators of Hankel type. Tohoku Math. J. 30, 163–171 (1978)
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The authors would like to thank the referee(s) for valuable suggestions and comments that led to the improvement of the paper.
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A part of the paper was completed during a scientific stay of the authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM), whose hospitality is gratefully appreciated. Xuan Thinh Duong was supported by the Australian Research Council through the Discovery Project 190100970. Ly Kim Ha was funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under Grant Number B2019-18-01.
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Dao, N.A., Duong, X.T. & Ha, L.K. Commutators of Cauchy–Fantappiè Type Integrals on Generalized Morrey Spaces on Complex Ellipsoids. J Geom Anal 31, 7538–7567 (2021). https://doi.org/10.1007/s12220-020-00561-5
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DOI: https://doi.org/10.1007/s12220-020-00561-5