#### Graduation Year

2005

#### Document Type

Dissertation

#### Degree

Ph.D.

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Vilmos Totik, Ph.D.

#### Committee Member

Mourad E. H. Ismail, Ph.D.

#### Committee Member

Evguenii A. Rakhmanov, Ph.D.

#### Committee Member

Boris Shekhtman, 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* ⊂ [0, 1] in the extended complex plane. We extend a result of V. Andrievskii which claims that if the Green function satisfies the Hölder1/2 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 *E* ⊂ [−1, 1]. In this case the maximal smoothness of the Green function is H¨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 a compact set in R^{d} , d > 2. We give a Wiener type characterization for the Hölder continuity of the Green function, thus extending a result of L. Carleson and V. Totik. The obtained density condition is necessary, and it is sufficient as well, provided *E* satisfies the cone condition. It is also shown that the Hölder condition for the Green function at a boundary point can be equivalently stated in terms of the equilibrium measure and the solution to the corresponding Dirichlet problem. The results solve a long standing open problem - raised by Maz’ja in the 1960’s - under the simple cone condition.

#### Scholar Commons Citation

Toókos, Ferenc, "Hölder Continuity of Green’s Functions" (2004). *USF Tampa Graduate Theses and Dissertations.*

https://digitalcommons.usf.edu/etd/1269