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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 $$\tilde{f} \in L^1_{p,w}(G)$$.)

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 31B15 Potentials and capacities, extremal length and related notions in higher dimensions
##### Keywords:
capacity; null sets; weighted Sobolev spaces
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