Revista Matemática Iberoamericana


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Volume 32, Issue 1, 2016, pp. 219–255
DOI: 10.4171/RMI/884

Published online: 2016-02-29

Fine properties of Newtonian functions and the Sobolev capacity on metric measure spaces

Lukáš Malý[1]

(1) University of Cincinnati, USA

Newtonian spaces generalize first-order Sobolev spaces to abstract metric measure spaces. In this paper, we study regularity of Newtonian functions based on quasi-Banach function lattices. Their (weak) quasi-continuity is established, assuming density of continuous functions. The corresponding Sobolev capacity is shown to be an outer capacity. Assuming sufficiently high integrability of upper gradients, Newtonian functions are shown to be (essentially) bounded and (H¨older) continuous. Particular focus is put on the borderline case when the degree of integrability equals the “dimension of the measure”. If Lipschitz functions are dense in a Newtonian space on a proper metric space, then locally Lipschitz functions are proven dense in the corresponding Newtonian space on open subsets, where no hypotheses (besides being open) are put on these sets.

Keywords: Newtonian space, Sobolev-type space, metric measure space, Banach function lattice, Sobolev capacity, quasi-continuity, outer capacity, locally Lipschitz function, continuity, doubling measure, Poincaré inequality

Malý Lukáš: Fine properties of Newtonian functions and the Sobolev capacity on metric measure spaces. Rev. Mat. Iberoamericana 32 (2016), 219-255. doi: 10.4171/RMI/884