On the weighted equivalence of open sets in $R^n$

V. A. Shlyk

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 4 (2014), pp. 47-52.

Résumé

Ahlfors and Beurling gave a characterization in terms of extremal distances of the removable singularities for the class of analytic functions with finite Dirichlet integral. Following Ahlfors and Beurling refer a relatively closed set $E$ contained in open set $G\subset R^n$ as an $NC_{p,w}$-set if $E$ do not affect the$(p,w)$-modulus $m_{p,w}(F_{0},F_{1},\Pi)$ for every coordinate rectangle $\Pi\subset G$. Dymchenko and Shlyk established that $NC_{p,w}$-sets are removable for the weighted Sobolev space $L^1_{p,w}(G)$. Observe that the idea to study removable sets of this type in $R^{n}$, $n\ge2$, in terms of rectangle is not new and for $w\equiv 1$ was considered by Hedberg, Yamamoto. In particular Hedberg gave the definition of null set $E\subset \Pi$ for a certain condenser capacity and showed that such set $E$ is removable for the class of real valued harmonic function $u$ with vanishing periods, $\int\left|\nabla u\right|^pdx\infty . $ Also remark that $NC_{p,w}$- sets were under investigation by Väisälä, Aseev and Sychev for $p=n$, $w\equiv $1; by Vodop'yanov and Gol'dshtein, $w\equiv $1. For more fully information about $NC_{p,w}$-sets, $w\equiv $1, we refer to the book by Gol'dshtein and Reshetnyak “Quasiconformal mappings and Sobolev Spaces”. Following Vodop'yanov and Gol'dshtein open sets $G_{1}$ and $G_2$ ($G_1\subset G_{2}$) will be called $(1,p,w)$-equivalent if the operator of restriction $\theta$: $L^1_{p,w}(G_2)\to L^1_{p,w}(G_1)$ is the isomorphism of the vector spaces $L^1_{p,w}(G_2)$ and $L^1_{p,w}(G_1)$. In the present paper we have established the criterion of $(1,p,w)$-equivalence of open sets in $R^n$: In order to open sets $G_1$ and $G_2$ $(G_1\subset G_2\subset R^n)$ be ($1,p,w)$-equivalent, necessary and sufficient that the set $G_2\setminus G_1$ be an $NC_{p,w}$-set in $G_2$. This result generalize the earlier criterion by Vodop'yanov and Gol'dstein and it's proof is used the definition of null-sets for the Muckenhoupt weight condenser module in Ahlfors–Beurling sense.

Détail

Keywords: modulus of curves family, capacity, Sobolev functions classes, Muckenhoupt weight.
Mots-clés : condenser

Bibliographie

[1] V.\;V. Aseev, A.\;V. Sychev, “On the removable sets for space quasiconformal mappings”, Siberian Mathematical Journal, 15:6 (1974), 1213–1227

[2] S.\;K. Vodopyanov, V.\;M. Goldshteyn, “Criteria of removability of sets for the spaces $L^1_p$, quasiconformal and quasiisometric mappings”, Siberian Mathematical Journal, 18:1 (1977), 48–68

[3] I.\;N. Demshin, Yu.\;V. Dymchenko, V.\;A. Shlyk, “The null-sets criteria for the weighted Sobolev spaces”, J. of Math. Sciences, 276 (2001), 52–82

[4] Yu.\;V. Dymchenko, V.\;A. Shlyk, “The sufficiency of families of polygonal curves in the moduli method and removable sets”, Siberian Mathematical Journal, 51:6 (2010), 1298–1315

[5] P.\;A. Pugach, V.\;A. Shlyk, “Generalized capacities and polyhedral surfaces”, J. of Math. Sciences, 383 (2010), 148–178

[6] G. Federer, Geometric measure theory, Nauka Publ., M., 1987, 760 pp.

[7] L. Ahlfors, A. Beurling, “Conformal invariants and functions-theoretic null-sets”, Acta Math., 83:1 (1950), 101–129

[8] B. Muckenhoupt, “The equivalence of two conditions for weight functions”, Studia Math., 49:2 (1974), 101–106

[9] J. Väisälä, “On the null-sets for extremal distances”, Ann. acad. Sci. Fenn. Ser. I. Math., 322 (1962), 1–12