Poincaré inequalities for the maximal function

Olli Saari

Published 2016-05-17Version 1

We study generalized Poincar\'e inequalities. We prove that if a function satisfies a suitable inequality of Poincar\'e type, then the Hardy-Littlewood maximal function also obeys a meaningful estimate of similar form. As a by-product, we get a unified approach to proving that the maximal operator is bounded on Sobolev, Lipschitz and BMO spaces. As another application, we show that the distributional derivatives of the maximal function of $u \in W^{1,1}(\mathbb{R}^{n})$ coincide with some functions in $L^{1,\infty}(\mathbb{R}^{n})$ outside a closed set of measure zero. The same property holds for $u \in {\rm BV}(\mathbb{R}^{n})$.