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

• Main Authors: Yong Hong, Qiliang Huang, Bicheng Yang, Jianquan Liao
• Format: Article
• Language: English
• Published: SpringerOpen 2017-12-01
• Series: Journal of Inequalities and Applications
• Subjects: Hilbert-type inequality; non-homogeneous kernel; sufficient and necessary conditions; best possible constant factor; bounded operator; operator norm
• Online Access: http://link.springer.com/article/10.1186/s13660-017-1592-8

Abstract For x = ( x 1 , … , x n ) ${x}= ( {x}_{1},\ldots, {x}_{{n}} )$ , u ( x ) = ( ∑ i = 1 n a i x i ρ ) 1 / ρ ${u} ( {x} ) = ( \sum_{{i}=1}^{{n}} {a}_{{i}} {x}_{{i}}^{\rho} )^{1/\rho}$ , v ( y ) = ( ∑ i = 1 n b i y i ρ ) 1 / ρ ${v} ( {y} ) = ( \sum_{{i}=1}^{{n}} {b}_{{i}} {y}_{{i}}^{\rho} )^{1/\rho}$ , 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 ) ) ${K} ( {u} ( {x} ),{v} ( {y} ) ) ={G} ( {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) )$ and the best possible constant factor are discussed. Furthermore, its application in the operator theory is considered.