Hölder quasicontinuity of Sobolev functions on metric spaces. (English) Zbl 1155.46306

Summary: We prove that every Sobolev function defined on a metric space coincides with a Hölder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Hölder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of J. Malý [Potential Anal. 2, No. 3, 249–254 (1993; Zbl 0803.46037)] to the Sobolev spaces on metric spaces [P. Hajłasz, Potential Anal. 5, No. 4, 403–415 (1996; Zbl 0859.46022)].


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
28A15 Abstract differentiation theory, differentiation of set functions
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