Czechoslovak Mathematical Journal, Vol. 71, No. 2, pp. 475-490, 2021
The boundedness of two classes of integral operators
Xin Wang, Ming-Sheng Liu
Received September 28, 2019. Published online December 18, 2020.
Abstract: The aim of this paper is to characterize the $L^p-L^q$ boundedness of two classes of integral operators from $L^p (\mathcal{U}, {\rm d} V_\alpha)$ to $L^q(\mathcal{U}, {\rm d} V_\beta)$ in terms of the parameters $a$, $b$, $c$, $p$, $q$ and $\alpha$, $\beta$, where $\mathcal{U}$ is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).
Keywords: integral operator; Siegel upper half-space; weighted $L^p$ space; boundedness
References: [1] O. Furdui: The Fock Space and Related Bergman Type Integral Operators: PhD. Thesis. Western Michigan University, Kalamazoo (2007). MR 2710271
[2] O. Furdui: On a class of integral operators. Integral Equations Oper. Theory 60 (2008), 469-483. DOI 10.1007/s00020-008-1572-y | MR 2390439 | Zbl 1157.47032
[3] O. Kures, K. Zhu: A class of integral operators on the unit ball of $\mathbb{C}^n$. Integral Equations Oper. Theory 56 (2006), 71-82. DOI 10.1007/s00020-005-1411-3 | MR 2256998 | Zbl 1109.47041
[4] C. Liu, Y. Liu, P. Hu, L. Zhou: Two classes of integral operators over the Siegel upper half-space. Complex Anal. Oper. Theory 13 (2019), 685-701. DOI 10.1007/s11785-018-0785-6 | MR 3940386 | Zbl 1421.32011
[5] M.-S. Liu: Biholomorphic convex mappings of order $\alpha$ on $B_p^n$. Complex Var. Elliptic Equ. 58 (2013), 899-908. DOI 10.1080/17476933.2011.603415 | MR 3170670 | Zbl 1277.32001
[6] M.-S. Liu, N. Li, Y. Yang: On the biholomorphic convex mappings of order alpha on $D_p^n$. Complex Anal. Oper. Theory 11 (2017), 243-260. DOI 10.1007/s11785-015-0528-x | MR 3605227 | Zbl 1364.32002
[7] M.-S. Liu, X.-M. Tang: Sufficient conditions for $\varepsilon$ quasi-convex mappings in a complex Banach space. Complex Var. Elliptic Equ. 58 (2013), 1273-1282. DOI 10.1080/17476933.2012.662224 | MR 3170698 | Zbl 1277.32003
[8] M.-S. Liu, F. Wu: Sharp inequalities of homogeneous expansions of almost starlike mappings of order alpha. Bull. Malays. Math. Sci. Soc. (2) 42 (2019), 133-151. DOI 10.1007/s40840-017-0472-1 | MR 3894620 | Zbl 1408.32003
[9] M.-S. Liu, F. Wu, Y. Yang: Sharp estimates of quasi-convex mappings of type B and order $\alpha$. Acta Math. Sci., Ser. B, Engl. Ed. 39 (2019), 1265-1276. DOI 10.1007/s10473-019-0506-x | MR 4068816
[10] R. Zhao: Generalization of Schur's test and its application to a class of integral operators on the unit ball of $\mathbb{C}^n$. Integral Equations Oper. Theory 82 (2015), 519-532. DOI 10.1007/s00020-014-2215-0 | MR 3369311 | Zbl 1319.47041
[11] L. Zhou: On the boundedness and the norm of a class of integral operators. Acta Math. Sci., Ser. B, Engl. Ed. 35 (2015), 1475-1482. DOI 10.1016/S0252-9602(15)30068-0 | MR 3413509 | Zbl 1349.47078
[12] K. Zhu: Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics 226. Springer, New York (2005). DOI 10.1007/0-387-27539-8 | MR 2115155 | Zbl 1067.32005
Affiliations: Xin Wang, Ming-Sheng Liu (corresponding author), School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, P. R. China, e-mail: 574149386@qq.com, liumsh65@163.com
