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A GENERALIZATION OF ORLICZ-SOBOLEV CAPACITY IN METRIC MEASURE SPACES


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

Issue:

SSRSMI, Number 2, Volume XIX

Section:

Volume 19, Number 2

Abstract:

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 N1,B(X). It was proved that B-capacity is an outer measure, which represents the correct gauge for distinguishing between two functions in N1,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 N1,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.

Keywords:

Banach function space, Sobolev space, Sobolev capacity, Choquet capacity.

Code [ID]:

SSRSMI200902V19S01A0026 [0003138]

Full paper:

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