Resumen de Removable sets for Newtonian Sobolev spaces and a characterization of p-path almost open sets

Anders Björn, Jana Björn, Panu Lahti

• 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 E Rn with zero Lebesgue measure is shown to be removable for W 1;p.Rn n E/ if and only if Rn n E supports a p-Poincaré inequality as a metric space. When p > 1, this recovers Koskela’s result (Ark. Mat. 37 (1999), 291–304), but for p D 1, as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces L1;p. To be able to include p D 1, we first study extensions of Newtonian Sobolev functions in the case p D 1 from a noncomplete space X to its completion Xy. In these results, p-path almost open sets play an important role, and we provide a characterization of them by means of p-path open, p-quasiopen and p-finely open sets. We also show that there are nonmeasurable ppath almost open subsets of Rn, n 2, provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with Lp-integrable upper gradients, about p-quasiopen, p-path open and p-finely open sets, and about Lebesgue points for N1;1-functions, to spaces that only satisfy local assumptions