Maximal Functions Measuring Smoothness

Abstract

The family of maximal operators measuring the local smoothness of Lp(ℝn) functions has been introduced in the work of A. Calderón, K.I. Oskolkov, and V.I. Kolyada. These maximal operators turned out to be useful in the solutions of a series of important problems of function theory (embedding theorems of Sobolev type, characterization of Sobolev spaces in terms of the metric and measure, quantitative estimates of Luzin property). Our survey is dedicated to the history of such operators and the main results related to them. We also consider properties and applications of such operators in the general context of metric spaces with a doubling measure. © Springer Science+Business Media, LLC 2013.