Abstract
We study the function Λm(X), 0<m<1, of compact setsX in ℝn, n≥2, defined as the distance in the spaceC m(X)≡lipm(X) from the function |x|2 to the subspaceH m (X) which is the closure inC m (X) of the class of functions harmonic in the neighborhood ofX (each function in its own neighborhood). We prove the equivalence of the conditions Λm(X)=0 andC m (X)=H m (X). We derive an estimate from above that depends only on the geometrical properties of the setX (on its volume).
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Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 372–382, September, 1997.
Translated by I. P. Zvyagin
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Gorokhov, Y.A. Approximation by harmonic functions in theC m-Norm and harmonicC m-capacity of compact sets in ℝn . Math Notes 62, 314–322 (1997). https://doi.org/10.1007/BF02360872
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DOI: https://doi.org/10.1007/BF02360872