Mathematics

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

V. A. Shlyk

Vladivostok Branch of Russian Customs Academy

Abstract: 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.

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

UDC: 517.51
BBK: 22.161.5