Hardy operator with variable limits on monotone functions
In the paper the authors study the Hardy operator \[ {\mathbb H}f(x)=\int_{a(x)}^{b(x)}f(y)u(y)\,dy \] with a non-negative weight function \(u\), restricted to the cone of non-increasing or non-decreasing non-negative functions on the semiaxis \({\mathbb R}^+:=[0,\infty)\). The border functions \(a(x)\) and \(b(x)\) are supposed to be continuous and strictly increasing on \({\mathbb R}^+\). Using the Sawyer criterion, the boundedness of the generalized Hardy operator with variable limits and the block-diagonal structure of \(\mathbb H\), weighted \(L^p - L^q\) inequalities for the operator \(\mathbb H\) on monotone functions are characterized.
scientific article; zbMATH DE number 2111350