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

نویسندگان

چکیده

We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling and Poincaré inequality.In particular, when restricted to Euclidean spaces, closed set $E \subset \mathbb{R}^n$ with zero Lebesgue is shown be $W^{1,p}(\mathbb{R}^n \setminus E)$ if only $\mathbb{R}^n E$ supports $p$-Poincaré inequality as space. When $p>1$, this recovers Koskela’s result (Ark. Mat. $\mathbf{37}$ (1999), 291–304), but $p=1$, well it seems new. also obtain corresponding characterization Dirichlet $L^{1,p}$. To able include we first extensions case $p=1$ from noncomplete space $X$ its completion $\hat X$. In these results, $p$-path almost open play an important role, provide them by means open, $p$-quasiopen $p$-finely sets. show that there are nonmeasurable subsets $\mathbb{R}^n$, $n \ge 2$, provided continuum hypothesis assumed true. Furthermore, extend earlier results about measurability $L^p$-integrable upper gradients, $p$-quasiopen, sets, points $N^{1,1}$-functions, satisfy local assumptions.

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ژورنال

عنوان ژورنال: Revista Matematica Iberoamericana

سال: 2023

ISSN: ['2235-0616', '0213-2230']

DOI: https://doi.org/10.4171/rmi/1419