On boundedness of a certain class of Hardy-Steklov type operators in Lebesgue spaces
Suppose that \(w\) and \(v\) are locally integrable non-negative weight functions. The paper under review studies \(L_p\)-\(L_q\) boundedness of the Hardy-Steklov type operator \(\mathcal{K}\) defined by \[ \mathcal{K}f(x)=w(x)\int_{a(x)}^{b(x)}k(x,y)f(y)v(y)dy, \] where the border functions \(a(x)\) and \(b(x)\) are differentiable and strictly increasing on \((0,\infty)\), \(a(0)=b(0)=0\), \(a(x)<b(x)\) for \(x\in (0,\infty)\), \(a(\infty)=b(\infty)=\infty\), and the kernel \(k(x,y)\) is strictly positive on the set \(\{(x,y):x>0,a(x)<y<b(x)\}\) and satisfies at least one of two generalized Oinarov's conditions \(\mathcal{O}_b\) and \(\mathcal{O}_a\).
scientific article; zbMATH DE number 5702398