Generalized Hardy operators and normalizing measures.
Let \(v\) be a non-negative weight on \([0,\infty)\) and let \((S, \sigma)\) be a measure space. Suppose that \(a\) and \(b\) are non-negative \(\sigma\)-measurable functions on \(S\) such that \(a(s) \leq b(s)\) for all \(s\in S\). Moreover, let \(k: S \times [0,\infty) \rightarrow [0,\infty)\) satisfy \[ C^{-1} k(s,y) \leq k(s,t) + k(t,y) \leq C k (s,y) \quad\text{if} \quad y \leq t \leq s, \] and \[ \sup \{k(t,y)\: z \leq y \leq b(t)<x\} = \sup\{k(t,y)\: z\leq y \leq b(t) \leq x\} \quad\text{if} \quad 0<z<x. \] The aim of the paper is to find necessary and sufficient conditions on the weight \(v\) and the measure \(\sigma\) which guarantee that the operator \[ (Kf) (s) = \int^{b(s)}_{a(s)} k(s,y) f(y)\,dy, \quad s\in S, \] is bounded from \(L^p_v [0,\infty)\) to \(L^q_{\sigma}(S)\) provided that \(1<p\leq q < \infty\) or \(1<q<p<\infty\). To this end, the authors introduced the concepts of a normalizing measure and a~nearly block diagonal decomposition of the operator \(K\). The results are applied to get estimates for the remainder in a Taylor approximation.
scientific article; zbMATH DE number 1902233