Shlyk V.A. On the weighted equivalence of open sets in Rn

 
 
Shlyk Vladimir Alеksееvich
 
Doctor of Physical and mathematical Sciences, professor, Department of Informatics
and Customs Informatics Technologies, Vladivostok Branch of Russian Customs Academy
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Strelkovaya st., 16-v, 690034 Vladivostok, Russian Federation
 
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⊂Rn as an NCp,w-set if E do not affect the(p,w)-modulus mp,w(F0,F1,Π) for every coordinate rectangle Π⊂G.
Dymchenko and Shlyk established that NCp,w-sets are removable for the weighted Sobolev space L1p,w(G). Observe that the idea to study removable sets of this type in Rn, n ≥ 2, in terms of rectangle is not new and for w ≡ 1 was considered by Hedberg, Yamamoto. In particular Hedberg gave the definition of null set E⊂Π 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, ∫︀|∇u|pdx<∞. Also remark that NCp,w- sets were under investigation by Väisälä, Aseev and Sychev for p = n, w ≡1; by Vodop’yanov and Gol’dshtein, w ≡1. For more fully information about NCp,w-sets, w ≡1, we refer to the book by Gol’dshtein and Reshetnyak “Quasiconformal mappings and Sobolev Spaces”. Following Vodop’yanov and Gol’dshtein open sets G1and G2(G1⊂G2) will be called (1,p,w)-equivalent if the operator of restriction θ: L1p,w(G2) → L1p,w(G1is the isomorphism of the vector spaces L1p,w(G2) and L1p,w(G1).
In the present paper we have established the criterion of (1,p,w)-equivalence of open sets in Rn: In order to open sets G1 and G2 (G1⊂G2⊂Rn) be (1,p,w)-equivalent, necessary and sufficient that the set G2∖Gbe an NCp,w-set in G2. 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.
 
Key words: modulus of curves family, condenser, capacity, Sobolev functions classes, Muckenhoupt weight.

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On the Weighted Equivalence of Open Sets in Rn by Shlyk V.A. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №4 (23) 2014 pp. 47-52
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