Addendum to “Symmetrization, symmetric stable processes, and Riesz capacities”

Abstract

In [1] we proved the following theorem. Theorem 1. Let α ∈ (0, 2). Let K be a compact set in R with mn(K) = V > 0. Let BV be a ball in R with mn(BV ) = V . Then Cα(K) ≥ Cα(BV ). Here mn(K) is the n-dimensional Lebesgue measure and Cα is the α-capacity defined by M. Riesz (see e.g. [2]). I, as well as several other people who work on such symmetrization results in potential theory, thought that this was an open problem. T. Watanabe informed me that in his paper [3] from 1983 he proved an isoperimetric inequality (Theorem 1) for certain Levy processes that include the symmetric stable processes. Moreover, his Theorem 2 contains an equality statement for the isoperimetric inequality. In the introduction of [3], Watanabe points out that his Theorems 1 and 2 hold for Riesz capacities, i.e. Theorem 1 above follows from Theorem 1 in [3]. The proof in [3] uses Fukushima’s theory of Dirichlet forms and is completely different from the proof of Theorem 1 in [1].