Minimally thin sets below a function graph

Abstract

A set E is minimally thin at a boundary point, ξ, if the Martin kernel with pole at ξ does not coincide with its balayage on E. Or in a probabilistic language: There is a non-zero probability that a Brownian motion that is conditioned to exit at ξ will avoid the set E. We will consider a special class of sets E, namely sets in the upper half-space that lies between the graph of a function and the boundary of the half-space. Brelot and Doob gave in 1963 an integral criterion for positive non-decreasing functions for minimal thinness of E. In 1991 Gardiner showed that the same criterion holds for the class of Lipschitz continuous functions. We will generalize these results to the class self-controlled functions, which is similar to the Beurling slow varying class of functions.

Keywords:

• Self-controlled
• Beurling slow varying
• Minimal thinness
• Whitney decomposition
• 2000 Mathematics Subject Classifications: 30C85
• 31A15
• 31B05
• 31B15
• 31C35

Acknowledgements

This paper is dedicated to the memory of my teacher Matts Essén. He was the most caring teacher you could wish for. I am most grateful to have been one of his students and friends. The present study was originally inspired by a Wiener-type criterion for minimal thinness (using a Whitney decomposition) which was introduced by Matts Essén in [9].

Notes

E-mail: [email protected]

Additional information

Notes on contributors

Torbjörn Lundh *

E-mail: [email protected]