×

Regularity of the fractional maximal function. (English) Zbl 1021.42009

The purpose of this work is to show that the fractional maximal operator has somewhat unexpected regularity properties. Our main result shows that the fractional maximal operator maps \(L^p\)-spaces boundedly into certain first-order Sobolev spaces. We also prove that the fractional maximal operator preserves first-order Sobolev spaces. This extends known results for the Hardy-Littlewood maximal operator.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDFBibTeX XMLCite
Full Text: DOI