The highest smoothness of the Green function implies the highest density of a set

Abstract

Abstract. We investigate local properties of the Green function of the complement of a compact set EC [O, 1] with respect to the extended complex plane. We demonstrate that if the Green function satisfies the 89 condition locally at the origin, then the density of E at 0, in terms of logarithmic capacity, is the same as that of the whole interval [0, 1]. Definitions and main resultsLet EC [0, 1] be a compact set with positive (logarithmic) capacity cap(E)>0.We consider E as a set in the complex plane C and use notions of potential theory in the plane (see [4] and [51).Let ft:=C\E, where C:-{oc}UC is the extended complex plane. Denote by ga(z)=gf~(z, oc), zEgt, the Green function of ft with pole at oc. In what follows we assume that 0 is a regular point of E, i.e., 9a(z) extends contimtously to 0 and g~(0)=0The monotonieity of the Green function yieldsthat is, if E has the "highest density" at 0, then ga has the "highest smoothness" at the origin. In particular g~( ~)_>gc\E0,11(-,-)> 89 0<~<1.In this regard, we would like to explore properties of E whose Green function has the "highest smoothness" at 0, that is, of E conforming to the following condition g~(z)<clz] ~-/2, c=const>O, zcC,

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