AN EXTENSION OF CHEEGER DIFFERENTIAL OPERATOR FROM LIPSCHITZ FUNCTIONS TO ORLICZ-SOBOLEV FUNCTIONS ON METRIC
MEASURE SPACES
MARCELINA MOCANU Department of Mathematics, Informatics and Education Sciences,
Faculty of Sciences, "Vasile Alecsandri" University of Bacău, 157 Calea Mărăşeşti, 600115 Bacău, ROMANIA, e-mail: mmocanu@ub.ro
We introduce two types of Orlicz-Sobolev spaces on a metric measure space. One space is the completion of locally Lipschitz functions in a norm of Orlicz-Sobolev type involving an abstract differentiation operator and the other space is de_ned via an Orlicz-Poincaré inequality. We prove that these spaces agree and are reflexive provided that the measure is doubling and the Young function defining the underlying Orlicz space is doubling, together with its complementary function. In the case where the Young func-tion is a power function with exponent greater than one, we recover some results of Fran-chi, Hajłasz and Koskela (1999).
