×

Singular integral operators with Bergman-Besov kernels on the ball. (English) Zbl 07093661

Summary: We completely characterize in terms of the six parameters involved the boundedness of all standard weighted integral operators induced by Bergman-Besov kernels acting between different Lebesgue classes with standard weights on the unit ball of \({\mathbb{C}}^N\). The integral operators generalize the Bergman-Besov projections. To find the necessary conditions for boundedness, we employ a new versatile method that depends on precise imbedding and inclusion relations among various holomorphic function spaces. The sufficiency proofs are by Schur tests or integral inequalities.

MSC:

47B34 Kernel operators
47G10 Integral operators
32A55 Singular integrals of functions in several complex variables
45P05 Integral operators
46E15 Banach spaces of continuous, differentiable or analytic functions
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
32A36 Bergman spaces of functions in several complex variables
30H25 Besov spaces and \(Q_p\)-spaces
30H20 Bergman spaces and Fock spaces

Software:

DLMF
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beatrous, F., Burbea, J.: Holomorphic Sobolev spaces on the ball. Diss. Math. 276, 57 (1989) · Zbl 0691.46024
[2] Cheng, G., Hou, X., Liu, C.: The singular integral operator induced by Drury-Arveson kernel. Complex Anal. Oper. Theory 12, 917-929 (2018) · Zbl 06873057
[3] Cheng, G., Fang, X., Wang, Z., Yu, J.: The hyper-singular cousin of the Bergman projection. Trans. Am. Math. Soc. 369, 8643-8662 (2017) · Zbl 1476.30170
[4] Čučković, Ž., McNeal, J.D.: Special Toeplitz operators on strongly pseudoconvex domains. Rev. Mat. Iberoam. 22, 851-866 (2006) · Zbl 1124.32004
[5] Doğan, Ö.F., Üreyen, A.E.: Weighted harmonic Bloch spaces on the ball. Complex Anal. Oper. Theory 12, 1143-1177 (2018) · Zbl 1395.31002
[6] Faraut, J., Korányi, A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88, 64-89 (1990) · Zbl 0718.32026
[7] Forelli, F., Rudin, W.: Projections on spaces of holomorphic functions in balls. Indiana Univ. Math. J. 24, 593-602 (1974) · Zbl 0297.47041
[8] Gagliardo, E.: On integral transformations with positive kernel. Proc. Am. Math. Soc. 16, 429-434 (1965) · Zbl 0161.32404
[9] Hartz, M.: On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces. Canad. J. Math. 69, 54-106 (2017) · Zbl 1481.47111
[10] Kaptanoğlu, H.T.: Bergman projections on Besov spaces on balls. Ill. J. Math. 49, 385-403 (2005) · Zbl 1079.32004
[11] Kaptanoğlu, H.T.: Carleson measures for Besov spaces on the ball with applications. J. Funct. Anal. 250, 483-520 (2007) · Zbl 1135.46014
[12] Kaptanoğlu, H.T., Üreyen, A.E.: Precise inclusion relations among Bergman-Besov and Bloch-Lipschitz spaces and \[H^\infty H\]∞ on the unit ball of \[{\mathbb{C}}^N\] CN. Math. Nachr. 291, 2236-2251 (2018) · Zbl 1404.32008
[13] Okikiolu, G.O.: On inequalities for integral operators. Glasg. Math. J. 11, 126-133 (1970) · Zbl 0205.11604
[14] Okikiolu, G.O.: Aspects of the Theory of Bounded Integral Operators in \[L^p\] Lp-Spaces. Academic, London (1971) · Zbl 0219.44002
[15] Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge Univ, New York (2010) · Zbl 1198.00002
[16] Rudin, W.: Function Theory in the Unit Ball of \[{\mathbb{C}}^nCn\], Grundlehren Math. Wiss., vol. 241, Springer, New York (1980) · Zbl 0495.32001
[17] Zaharjuta, V.P., Judovič, V.I.: The general form of a linear functional in \[H_p^{\prime }\] Hp′. Uspekhi Mat. Nauk 19, 139-142 (1964)
[18] Zhao, R.: Generalization of Schur’s test and its application to a class of integral operators on the unit ball of \[{\mathbb{C}}^nCn\]. Integral Equ. Oper. Theory 82, 519-532 (2015) · Zbl 1319.47041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.