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. (English) Zbl 1386.26025
Summary: For \({x}= ( {x}_{1},\dots, {x}_{{n}} )\), \({u} ( {x} ) = ( \sum_{{i}=1}^{{n}} {a}_{{i}} {x}_{{i}}^{\rho} )^{1/\rho}\), \({v} ( {y} ) = ( \sum_{{i}=1}^{{n}} {b}_{{i}} {y}_{{i}}^{\rho} )^{1/\rho}\), 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 \({K} ( {u} ( {x} ),{v} ( {y} ) ) ={G} ( {u}^{\lambda_{1}} ( {x} ) {v}^{\lambda_{2}} (y) )\) and the best possible constant factor are discussed. Furthermore, its application in the operator theory is considered.
MSC:
| 26D15 | Inequalities for sums, series and integrals |
| 47A07 | Forms (bilinear, sesquilinear, multilinear) |
Keywords:
Hilbert-type inequality; non-homogeneous kernel; sufficient and necessary conditions; best possible constant factor; bounded operator; operator normReferences:
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