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Sharp self-improving properties of generalized Orlicz-Poincaré inequalities in connected metric measure spaces. (English) Zbl 1213.46030
Summary: We study the self-improving properties of generalized $$\Phi$$-Poincaré 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 $$\Phi$$-Poincaré inequality. Our results are optimal and generalize some of the results of A. Cianchi [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 23, No. 3, 575–608 (1996; Zbl 0877.46023); Indiana Univ. Math. J. 45, No. 1, 39–65 (1996; Zbl 0860.46022)], P. Hajłasz and P. Koskela [C. R. Acad. Sci., Paris, Sér. I 320, No. 10, 1211–1215 (1995; Zbl 0837.46024); Mem. Am. Math. Soc. 688 (2000; Zbl 0954.46022)], P. MacManus and C. Pérez [Trans. Am. Math. Soc. 354, No. 5, 1997–2012 (2002; Zbl 1032.46050)], Z. Balogh, K. Rogovin and T. Zürcher [J. Geom. Anal. 14, No. 3, 405–422 (2004; Zbl 1069.28001)] and E. M. Stein [Ann. Math. (2) 113, 383–385 (1981; Zbl 0531.46021)].

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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