Harmonic functions on locally compact groups of polynomial growth

Abstract

We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most k is finite dimensional, to the settings of locally compact groups equipped with measures with non-compact support.

Equivariant discretizations of diffusions and harmonic functions of bounded growth

“…(P1) the support of µ is all of Γ; (P2) µ is symmetric; (P3) µ has finite exponential moment (for some sufficiently small exponent). In particular, in the transient case, µ satisfies the properties required in the articles [19,20,21] of Meyerovitch, Perl, Tointon, and Yadin so that we may apply their results, using Theorem B.…”

“…We discuss H d (M, L) for d ≥ 1. Our strategy consists of combining results of Meyerovitch, Perl, Tointon, and Yadin [19,20,21] about µ-harmonic functions on groups and translating them using Theorem B. More detailed references will be given in the text.…”

“…for all d ≥ 1, where µ is a courteous probability measure on Γ in the sense of [20], c 1 < c 2 are positive constants, and r is the rank of the nilpotent subgroup N ⊆ Γ of finite index. Here we use [19,Corollary 1.9] and [21,Theorem 1.5] to pass from finitely supported, symmetric probability measures µ on Γ, whose support generates Γ, as assumed in [19,Corollary 1.12], to the more general class of courteous probability measures. This class includes the probability measures on Γ induced from LS-measures as used here, at least in the case where the LS-data are appropriately chosen and Γ does not contain Z or Z 2 as a subgroup of finite index.…”

Polynomially growing harmonic functions on connected groups

“…Proof. In [Per18] it is shown that for any CGLC group of polynomial growth, G, and any courteous µ with exponential tail on G, the dimension of HF k (G, µ) is finite for all k ≥ 1. (This is an extension of Kleiner's work [Kle10] to non-compactly-supported measures, and to connected CGLC groups.)…”