ACTA issues

## Smoothness of Green's functions and density of sets

F. Toókos

Acta Sci. Math. (Szeged) 71:1-2(2005), 117-146
5861/2009

 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.)