2004

Dissertation

Ph.D.

Degree Granting Department

Mathematics and Statistics

Major Professor

Vilmos Totik, Ph.D.

Keywords

Logarithmic capacity, Newtonian potential, Equilibrium measure, Boundary behavior, Wiener's criterion

Abstract

We investigate local properties of the Green function of the complement of a compact set\$E\$. First we consider the case \$E\subset [0,1]\$ in the extended complex plane. We extend a result of V. Andrievskii which claims that if the Green function satisfies the H\"older-\$1/2\$ 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]\$. We give an integral estimate on the density in terms of the Green function, which also provides a necessary condition for the optimal smoothness. Then we extend the results to the case \$Esubset [-1,1]. In this case the maximal smoothness of the Green function is "older-1 and a similar integral estimate and necessary condition hold as well.In the second part of the paper we consider the case when \$E\$ is acompact set in R, >2. We give a Wiener type characterization for the "older continuity of the Green function, thus extending a result of L.

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