A note on the equilibrium potential of certain Dirichlet spaces (Q679214)

Suppose we are given a Dirichlet space corresponding to a symmetric Levy process on \(\mathbb{R}^m\). It is well known that, for any compact subset \(E\) of \(\mathbb{R}^m\) with nonempty interior, the equilibrium measure of \(E\) with respect to the given Dirichlet space exists. This paper proves that the restriction of this equilibrium measure to the interior of \(E\) is absolutely continuous with respect to the Lebesgue measure and gives an explicit formula for its density. This result is an extension of H. Wallin's 1965 result on the equilibrium measure with respect to symmetric stable processes or Riesz potentials.