The necessary and sufficient conditions for the existence of a kind of Hilbert-type multiple integral inequality with the non-homogeneous kernel and its applications

• Published in: Journal of inequalities and applications Vol. 2017; no. 1; pp. 1 - 12
• Main Authors: Hong, Yong, Huang, Qiliang, Yang, Bicheng, Liao, Jianquan
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 28.12.2017; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; best possible constant factor; bounded operator; Hilbert-type inequality; Integrals; Mathematics; Mathematics and Statistics; non-homogeneous kernel; operator norm; Operators (mathematics); sufficient and necessary conditions; best possible constant factor; sufficient and necessary conditions; 47A07; operator norm; Hilbert-type inequality; non-homogeneous kernel; bounded operator; 26D15
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-017-1592-8

For x = ( x 1 , … , x n ) , u ( x ) = ( ∑ i = 1 n a i x i ρ ) 1 / ρ , v ( y ) = ( ∑ i = 1 n b i y i ρ ) 1 / ρ , by using the methods and techniques of real analysis, the sufficient and necessary conditions for the existence of the Hilbert-type multiple integral inequality with the kernel K ( u ( x ) , v ( y ) ) = G ( u λ 1 ( x ) v λ 2 ( y ) ) and the best possible constant factor are discussed. Furthermore, its application in the operator theory is considered.