Boundedness criteria for the Hardy-Steklov operator expressed in terms of a fairway function
Let \(s\in\mathbb R\) and \(g\), \(\varphi\), \(\psi\) be nonnegative functions on the half-axis. If there exists a function \(\sigma(x)\) with values between \(\varphi(x)\) and \(\psi(x)\) such that \[ \int_{\varphi(x)}^{\sigma(x)}g^{s}(x)\,dy = \int_{\sigma(x)}^{\psi(x)}g^{s}(x)\,dy, \quad x>0, \] it is called a fairway function. The Hardy-Steklov operator is defined on the functions on the half-axis as follows \[ ({\mathcal H}f)(x) = \int_{a(x)}^{b(x)}f(y)\,dy, \quad x>0. \] In this paper, using the method of fairway functions, two-sided estimates are given for the norms of these operators acting from one weighted \(L_{p}\)-space on the half-space into another such space. The estimates depend only on these spaces and the functions \(a(x), b(x)\).
scientific article; zbMATH DE number 6474462