A GENERALIZATION OF ORLICZ-SOBOLEV CAPACITY IN METRIC MEASURE SPACES

MARCELINA MOCANU University "Vasile Alecsandri" of BacÄu, Department of Mathematics and Informatics, Spiru Haret 8, BacÄu 600114, Romania, e-mail address: mmocanu@ub.ro

Given a Banach function space B and a metric measure space X, we investigate continuity and regularity properties of the B-capacity, that we introduced in [13] by means of a Sobolev-type space N^{1,B}(X). It was proved that B-capacity is an outer measure, which represents the correct gauge for distinguishing between two functions in N^{1,B}(X) [13] . In the case when B is reflexive we show that B-capacity is continuous on increasing sequences of arbitrary subsets of X. Assuming that B has absolutely continuous norm, that every function in B is dominated by a semicontinuous function in B and that continuous functions are dense in N^{1,B}(X), we prove that B-capacity is outer regular. As consequences of this outer regularity we obtain the continuity of B-capacity on decreasing sequences of compact subsets of X and the coincidence between the B-capacity and another usual capacity.