Kinnunen, Juha; Saksman, Eero Regularity of the fractional maximal function. (English) Zbl 1021.42009 Bull. Lond. Math. Soc. 35, No. 4, 529-535 (2003). 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. Reviewer: Juha Kinnunen (Helsinki) Cited in 1 ReviewCited in 92 Documents MSC: 42B25 Maximal functions, Littlewood-Paley theory 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:fractional maximal operator; regularity; Sobolev spaces; Hardy-Littlewood maximal operator PDFBibTeX XMLCite \textit{J. Kinnunen} and \textit{E. Saksman}, Bull. Lond. Math. Soc. 35, No. 4, 529--535 (2003; Zbl 1021.42009) Full Text: DOI