Maximal functions, Riesz potentials and Sobolev’s inequality in generalized Lebesgue spaces. (English) Zbl 1125.31001

Mizuta, Yoshihiro; Shimomura, Tetsu

Aikawa, Hiroaki (ed.) et al., Potential theory in Matsue. Selected papers of the international workshop on potential theory, Matsue, Japan, August 23–28, 2004. Tokyo: Mathematical Society of Japan (ISBN 4-931469-33-7/hbk). Advanced Studies in Pure Mathematics 44, 255-281 (2006).

The aim of this paper is to deal with the boundedness of maximal functions in Lebesgue spaces \(L^{p(\cdot)}\) with variable exponent \(p(\cdot)\). As an application of the boundedness of maximal functions the authors show Sobolev’s inequality for Riesz potentials with variable exponent.
For the entire collection see [Zbl 1102.31001].

Reviewer: Olof Svensson (Norrköping)

MSC:

31B15	Potentials and capacities, extremal length and related notions in higher dimensions
42B25	Maximal functions, Littlewood-Paley theory
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

variable exponent Lebesgue spaces; boundedness of maximal function