ACTA issues

Smoothness of Green's functions and density of sets

F. Toókos

Acta Sci. Math. (Szeged) 71:1-2(2005), 117-146

Abstract. We investigate local properties of the Green function of the complement of a compact set $E\subset[0,1]$ with respect to the extended complex plane. We extend results of V. Andrievskii, L. Carleson and V. Totik which claim that the Green function satisfies the $1/2$-Hölder condition locally at the origin if and only if the density of $E$ at $0$, in terms of logarithmic capacity, is the same as that of the whole interval $[0,1]$. We give an integral estimate on the density in terms of the Green function and extend the results to the case $E\subset[-1,1]$.

AMS Subject Classification (1991): 30C10, 30C15, 41A10

Keyword(s): Green function, Equilibrium measure, Conformal invariants, Compact sets

Received September 27, 2004, and in revised form November 19, 2004. (Registered under 5861/2009.)