Title: Sharp self-improving properties of generalized Orlicz-Poincare inequalities in connected metric measure spaces

Authors: Toni Heikkinen

Issue: Volume 59 (2010), Issue 3, 957-986

Abstract: We study the self-improving properties of generalized $Phi$-Poincar\'e inequalities in connected metric spaces equipped with a doubling measure. As a consequence we obtain results concerning integrability, continuity and differentiability of Orlicz-Sobolev functions on spaces supporting a $\Phi$-Poincar\'e inequality. Our results are optimal and generalize some of the results of Cianchi (A. Cianchi, \emph{Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems}, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) \textbf{23} (1996), 575--608; A. Cianchi, \emph{A sharp embedding theorem for Orlicz-Sobolev spaces}, Indiana Univ. Math. J. \textbf{45} (1996), 39--65); Haj\l asz and Koskela (P. Haj\l asz and P. Koskela, \emph{Sobolev meets Poincar\'e}, C.R. Acad. Sci. Paris S\'er. I Math. \textbf{320} (1995), 1211--1215; see also their article by the same title in Mem. Amer. Math. Soc. \textbf{145} (2000), Number 688, x + 101 pp.); P. MacManus and C. P'erez, \emph{Trudinger inequalities without derivatives}, Trans. Amer. Math. Soc. \textbf{354} (2002), 1997--2012; Z. Balogh, K. Rogovin and T. Z\"urcher, \emph{The Stepanov differentiability theorem in metric measure spaces}}, J. Geom. Anal. \textbf{14} (2004), 405--422; and E.M. Stein \emph{Editor's note: The differentiability of functions in $\mathbb{R}^n$}, Annals of Math (2) \textbf{113} (1981), 383--385).