Poincaré inequalities for the maximal function

Abstract

We study generalized Poincaré inequalities. We prove that if a function satisfies a suitable inequality of Poincaré 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.

19 pages, 1 figure, a change in the abstract and a mistake removed. The application to $W^{1,1}$ reproduces the known results but does not improve them. All theorems remain as they were

Cited by in corpus (3)

• Regularity of maximal functions on Hardy-Sobolev spaces
• Regularity of maximal operators: recent progress and some open problems
• Maximal operators on spaces BMO and BLO