Geodesic
Removable sets for Sobolev spaces with Muckenhoupt $A_1$-weight
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 136-159.

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Let $\Omega$ be an open set in $R^n$, $n\ge2$, and $E$ be a relatively closed subset of $\Omega$. In this paper we obtain a criterion of equality $L^1_{1,\omega}(\Omega\setminus E)=L^1_{1,\omega}(\Omega)$ in terms of $E$ as an $NC_{1,\omega}$-set in $\Omega$ with $A_1$-weight $\omega$. In addition, we establish exact characterizations of $NC_{1,\omega}$-sets in terms of $NED_{1,\omega}$-sets and of the $(1,\omega)$-girth condition. In the case $\omega\equiv1$, these results complete the studies of Vodop'yanov and Gol'dstein on removable sets for $L^1_p(\Omega)$, $p\in(1,+\infty)$.
Keywords: Sobolev space, capacity and modulus of condenser, Muckenhoupt weight, removable set. 
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V. A. Shlyk. Removable sets for Sobolev spaces with Muckenhoupt $A_1$-weight. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 136-159. http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a40/

[1] L. Ahlfors, A. Beurling, “Conformal invariants and function-theoretic null-sets”, Acta Math., 83 (1950), 101–129 | DOI | MR | Zbl

[2] G.P. Akilov, B.M. Makarov, V.P. Khavin, An Elementary introduction to integration theory, Leningrad Univ., Leningrad, 1969

[3] V. Aseev, “On a modulus property”, Sov. Math., Dokl., 12 (1971), 1409–1411 | MR | Zbl

[4] V. Aseev, “NED sets on a hyperplane”, Sib. Math. J., 50:5 (2009), 760–775 | DOI | MR | Zbl

[5] S.-K. Chua, “Extension theorems on weighted Sobolev spaces”, Indiana Univ. Math. J., 41:4 (1992), 1027–1076 | DOI | MR | Zbl

[6] Y.V. Dymchenko, V.A. Shlyk, “Sufficiency of broken lines in the modulus method and removable sets”, Sib. Math. J., 51:6 (2010), 1028–1042 | DOI | MR | Zbl

[7] H. Federer, Geometric measure theory, Springer, Berlin etc, 1969 | MR | Zbl

[8] V. Gol'dshtejn, Yu. Reshetnyak, Quasiconformal mappings and Sobolev spaces, Mathematics and its Applications, Soviet Series, 54, Kluwer Academic Publishers, Dordrecht etc., 1990 | MR | Zbl

[9] J. Hesse, “A $p$-extremal length and $p$-capacity equality”, Ark. Mat., 13:1 (1975), 131–144 | DOI | MR | Zbl

[10] E. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, 14, AMS, Providence, 2001 | DOI | MR | Zbl

[11] V.G. Mazya, Sobolev spaces, Springer-Verlag, Berlin etc, 1985 | MR | Zbl

[12] B. Muckenhoupt, “Weighted norm unequalities for the Hardy maximal functions”, Trans. Am. Math. Soc., 165 (1972), 207–226 | DOI | MR | Zbl

[13] M. Ohtsuka, Extremal length and precise functions, GAKUTO International Series, Mathematical Sciences and Applications, 19, Gakkōtosho, Tokyo, 2003 | MR | Zbl

[14] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1987 | MR | Zbl

[15] B.O. Turesson, Nonlinear potential theory and weighted Sobolev spaces, Lecture Notes in Mathematics, 1736, Springer, Berlin, 2000 | DOI | MR | Zbl

[16] J. Väisälä, “On the null-sets for extremal distances”, Ann. Acad. Sci. Fenn., Ser. A I, 322 (1962) | MR | Zbl

[17] J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, 229, Springer-Verlag, Berlin etc., 1971 | MR | Zbl

[18] S.K. Vodop'yanov, V.M. Gol'dshtein, “Criteria for the removability of sets in spaces of {$L_p^1$} quasiconformal and quasi-isometric mappings”, Sib. Math. J., 18:1 (1977), 35–50 | DOI | Zbl