Removable sets for Newtonian Sobolev spaces and a characterization of -path almost open sets
Anders Björn
Linköping University, SwedenJana Björn
Linköping University, SwedenPanu Lahti
Chinese Academy of Sciences, Beijing, China
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Abstract
We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincaré inequality.In particular, when restricted to Euclidean spaces, a closed set with zero Lebesgue measure is shown to be removable for if and only if supports a -Poincaré inequality as a metric space. When , this recovers Koskela’s result (Ark. Mat. (1999), 291–304), but for , as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces . To be able to include , we first study extensions of Newtonian Sobolev functions in the case from a noncomplete space to its completion . In these results, -path almost open sets play an important role, and we provide a characterization of them by means of -path open, -quasiopen and -finely open sets. We also show that there are nonmeasurable -path almost open subsets of , , provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with -integrable upper gradients, about -quasiopen, -path open and -finely open sets, and about Lebesgue points for -functions, to spaces that only satisfy local assumptions.
Cite this article
Anders Björn, Jana Björn, Panu Lahti, Removable sets for Newtonian Sobolev spaces and a characterization of -path almost open sets. Rev. Mat. Iberoam. 39 (2023), no. 3, pp. 1143–1180
DOI 10.4171/RMI/1419