Shlyk V.A. On the weighted equivalence of open sets in Rn
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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
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.
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(G1) is 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∖G1 be 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.
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∖G1 be 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.
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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