#### Title

Holder continuity of green's functions

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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