Compactness of Hardy--Steklov operator. (Q1419742)

Let \(a\) and \(b\) denote strictly increasing functions on \([0,\infty]\) such that \(a(0) = b(0) = 0\), \(a(\infty) = b(\infty) = \infty\) and \(a(x) < b(x)\) for \(x\in (0,\infty)\). The aim of the paper is to characterize the compactness of the Hardy-Steklov operator \((Tf)(x) = \int^{b(x)}_{a(x)} f(t)\,dt\) between weighted Lebesgue spaces \(L^p(u)\) and \(L^q(v)\) provided that either \(1<p\leq q < \infty\) or \(0<q<p<\infty\), \(p>1\).