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# Minimally thin sets below a function graph

Torbjörn Lundh (Institutionen för matematik)
Complex Variables. Theory and Application Vol. 49 (2004), 7-9, p. 639-645.

A set $E$ is minimally thin at a boundary point, $\xi$, if the Martin kernel with pole at $\xi$ does not coincide with its balayage on $E$. %it is not big enough to lift the Poisson kernel''. Or in a probabilistic language: There is a non-zero probability that a Brownian motion that is conditioned to exit at $\xi$ 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. %(so called epigraphs). Brelot and Doob gave in 1963 an integral criterion for positive non-decreasing functions for minimally 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 {\em self-controlled} functions, which is similar to the {\em Beurling slow varying} class of functions.

Nyckelord: Self-controlled, Beurling slow varying, minimal thinness, Whitney decomposition

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

CPL Pubid: 23843

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# Institutioner (Chalmers)

Institutionen för matematik (2002-2004)