Two criteria of Wiener type for minimally thin sets and rarefied sets in a cylinder

Abstract

We shall give two criteria of Wiener type which characterize minimally thin sets and rarefied sets in a cylinder. We shall also show that a positive superharmonic function on a cylinder behaves regularly outside a rarefied set in a cylinder.

On the cylindrical Green's function for representation theory and its applications

“…There exists a measure τ on C n (Γ) such that supp τ ⊂ B j and τ (B j ) = 1. So we have for any W ∈ B jˆH which together with Theorem 1 in [9], gives that E is not rarefied at +∞ in H(Γ).…”

“…On the other hand, the term "Green's function" has entered the vocabulary of probabilists in a way that may seem initially unrelated, namely as a measure the expected number of times that a discrete process visits a point, or the expected amount of time that a continuous process spends at a point. The reason that these two different notions have garnered the same name was discovered by Miyamoto and his collaborators in [10,9], who showed that in many cases in R n these two notions coincide, with L the Laplacian and the process in question Markov chains.…”

“…We keep using the same notations appearing in [9]. Let B(V, r) denote the open ball with center at V and radius r in R n , where V = (X, y) ∈ R n and r > 0.…”