Null-sets criteria for weighted Sobolev spaces.

*(English. Russian original)*Zbl 1089.46022
J. Math. Sci., New York 118, No. 1, 4760-4777 (2003); translation from Zap. Nauchn. Semin. POMI 276, 52-82 (2001).

The authors give functional, capacity and metric characterizations of null sets on weighted Sobolev spaces \(L^1_{p,w}(G)\), with \(G\) an open subset of the \(n\)-dimensional Euclidean space. The weights are the usual Muckenhoupt \(A_p\) weights, and the norm on \(L^1_{p,w}(G)\) is given by
\[
\int_G | \nabla u| ^p w \, dx.
\]
By a theorem of Fabes, Kenig and Serapioni, it follows that a function in \(C_0^{\infty}(G)\) with finite norm is actually in a suitable \(L^q\) space with weight \(w\).

A condenser is a system \((E_0, E_1, G) = (G)\), where \(E_0, E_1 \subset \overline {G}\) are disjoint nonempty compact sets. For a condenser \((E_0, E_1, G)\), the \(p\)-capacity with weight \(w \in A_p\) is defined by the formula \[ C_{p,w}(G) = C_{p,w}(E_0, E_1, G) = \inf \int_G | \nabla u| ^p w \, dx. \] The infimum is taken over a suitable family of admissible functions, and \(E_{p,w}(G)\) denotes the class of extremal functions (admissible functions that assume the infimum). A set \(E\), closed with respect to \(G,\) is called an \(NC_{p,w}\)-set if \[ C_{p,w}(E_0, E_1, G \setminus E) = C_{p,w}(E_0, E_1, G) \] for any two disjoint smooth compact sets \(E_0, E_1 \subseteq G \setminus E\).

Among the authors’ results are (in increasing order of strength): \(NC_{p,w}\)-sets have no interior, compact subsets of an \(NC_{p,w}\)-set are \(NC_{p,w}\)-sets, an \(NC_{p,w}\)-set is of Lebesgue measure zero, and an \(NC_{p,w}\)-set is of dimension less than or equal to \(n - 2\). They also characterize \(NC_{p,w}\)-sets in terms of \(\epsilon\)-girth with respect to lines parallel to the coordinate axes and use that to derive an analog of the Ahlfors-Beurling theorem on \(NED\)-sets. They prove that \(E\) is a removable set for \(L^1_{p,w}(G)\) if and only if \(E\) is a \(NC_{p,w}\)-set. (A set \(E\) is removable for \(L^1_{p,w}(G)\) if and only if every \(f \in L^1_{p,w}(G \setminus E)\) can be extended to a function \( \tilde{f} \in L^1_{p,w}(G)\).)

A condenser is a system \((E_0, E_1, G) = (G)\), where \(E_0, E_1 \subset \overline {G}\) are disjoint nonempty compact sets. For a condenser \((E_0, E_1, G)\), the \(p\)-capacity with weight \(w \in A_p\) is defined by the formula \[ C_{p,w}(G) = C_{p,w}(E_0, E_1, G) = \inf \int_G | \nabla u| ^p w \, dx. \] The infimum is taken over a suitable family of admissible functions, and \(E_{p,w}(G)\) denotes the class of extremal functions (admissible functions that assume the infimum). A set \(E\), closed with respect to \(G,\) is called an \(NC_{p,w}\)-set if \[ C_{p,w}(E_0, E_1, G \setminus E) = C_{p,w}(E_0, E_1, G) \] for any two disjoint smooth compact sets \(E_0, E_1 \subseteq G \setminus E\).

Among the authors’ results are (in increasing order of strength): \(NC_{p,w}\)-sets have no interior, compact subsets of an \(NC_{p,w}\)-set are \(NC_{p,w}\)-sets, an \(NC_{p,w}\)-set is of Lebesgue measure zero, and an \(NC_{p,w}\)-set is of dimension less than or equal to \(n - 2\). They also characterize \(NC_{p,w}\)-sets in terms of \(\epsilon\)-girth with respect to lines parallel to the coordinate axes and use that to derive an analog of the Ahlfors-Beurling theorem on \(NED\)-sets. They prove that \(E\) is a removable set for \(L^1_{p,w}(G)\) if and only if \(E\) is a \(NC_{p,w}\)-set. (A set \(E\) is removable for \(L^1_{p,w}(G)\) if and only if every \(f \in L^1_{p,w}(G \setminus E)\) can be extended to a function \( \tilde{f} \in L^1_{p,w}(G)\).)

Reviewer: Raymond Johnson (College Park)