ON WEIGHTED BILINEAR INEQUALITY WITH A KERNEL

Abstract

Introducing a kernel into integral bilinear Hardy-type operators represents a new direction in operator theory. In the general case, since the criterion for the validity of an integral Hardy-type inequality with a kernel in a weighted Lebesgue space is not defined, various conditions are determined on the kernel, and wider results are obtained than in the case without a kernel. This paper examines integral bilinear Hardy-type operators involving the Hardy and Hardy-Volterra operators in weighted Lebesgue spaces. We obtained criteria for the validity of bilinear integral Hardy-type inequalities with kernels in the classes   O₁⁺ and O₁^- for the parameter ranges 1<max{p,r}< q<∞, 1<min{p,r}<q<max{p,r}<∞ and 1<q<min{p,r}<∞. An iteration technique is applied to characterize the boundedness of certain multilinear operators, reducing the problem to a linear operator case. This study contributes to the theory of integral operators in functional analysis and can be used in finding solutions to differential equations.