A class of bounded operators on Sobolev spaces
A result of J. Kinnunen states that the Hardy-Littlewood maximal operator is bounded on \(H^1_p({\mathbb R}^n)\) whenever \(1< p < \infty\). The main purpose of this paper is to extend this result by showing that the Hardy-Littlewood maximal operator is actually bounded on \(H^s_p({\mathbb R}^n)\) whenever \(1< p < \infty\) and \(0<s\leq 1\). For this purpose, the author introduces a class \(C\) of nonlinear bounded operators on \(L^r({\mathbb R}^n)\) for some \(1< r< \infty\) satisfying some additional conditions, and shows that each \(T\in C\) defines a bounded operator on the Sobolev spaces \(H^s_p({\mathbb R}^n)\) whenever \(1< p < \infty\) and \(0<s\leq 1\). The author obtains the main conclusion by showing that the Hardy-Littlewood maximal operator is in fact in \(C\).
scientific article; zbMATH DE number 2101539