A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applications

• Published in: Journal of inequalities and applications Vol. 2020; no. 1; pp. 1 - 13
• Main Authors: Hong, Yong, Liao, Jianquan, Yang, Bicheng, Chen, Qiang
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 13.05.2020; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Bounded operator; Generalized homogeneous kernel; Hilbert-type multiple integral inequality; Integrals; Kernels; Mathematics; Mathematics and Statistics; Necessary and sufficient condition; Operator norm; Operators (mathematics); The best constant factor; Necessary and sufficient condition; Generalized homogeneous kernel; Hilbert-type multiple integral inequality; Operator norm; Bounded operator; The best constant factor; 26D15
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-020-02401-0

Let x = ( x 1 , x 2 , … , x n ) , and let K ( u ( x ) , v ( y ) ) satisfy u ( r x ) = r u ( x ) , v ( r y ) = r v ( y ) , K ( r u , v ) = r λ λ 1 K ( u , r − λ 1 λ 2 v ) , and K ( u , r v ) = r λ λ 2 K ( r − λ 2 λ 1 u , v ) . In this paper, we obtain a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with kernel K ( u ( x ) , v ( y ) ) and discuss its applications in the theory of operators.