The equivalent conditions for norm of a Hilbert-type integral operator with a combination kernel and its applications

Abstract

Introducing adaptation parameters $\sigma ,{\sigma }_1$, formal parameters ${\lambda }_i(i=1,2,3,4),\kappa ,\tau$, and type parameters $\mu ,\nu$, the integration operator is defined as $T:L_p^{p(1-\mu \hat{\sigma })-1}\left(R_{+}\right)\to L_p^{p\nu \hat{\sigma }-1}\left(R_{+}\right)$, $Tf\left(y\right)={\int }_{R_{+}}\frac{e^{{\lambda }_1{x}^{\mu }{y}^{\nu }}+\kappa e^{-{\lambda }_2{x}^{\mu }{y}^{\nu }}}{e^{{\lambda }_3{x}^{\mu }{y}^{\nu }}+\tau e^{-{\lambda }_4{x}^{\mu }{y}^{\nu }}}f\left(x\right)dx,y\in R_{+}$. Using the weight function method, a general Hilbert-type integral inequality is obtained, thereby proving the boundedness of the operator. The constant factor of the general Hilbert-type inequality is the best possible if and only if the adaptation parameters satisfy $\sigma ={\sigma }_1$. From this, the formula for calculating the operator norm is obtained. In terms of application, some results from the references have been consolidated by discussing the combination of formal parameters and type parameters, and many new operator inequalities of different types and forms have been derived.