Harmonic Function Spaces on Groups

Abstract

A class of harmonic function spaces is introduced and studied,

Amenability and harmonic $L^p$-functions on hypergroups

“…In the case where σ is adapted and K is compact, we show that H p σ (K) = C1 for all 1 ≤ p ≤ ∞. These extend the results for the group case in [5].…”

“…We commence with the following lemma, whose proof is similar to those given in [5]. For completeness, we present the argument here.…”

“…It is a well-known result of [4] that if G is abelian, then the only bounded continuous σ-harmonic functions are constant functions when the support of σ generates a dense subgroup of G. Bounded harmonic functions have been investigated by several authors for various kinds of groups, e.g., nilpotent groups and compact groups ( [7]- [8], [10]- [12]). Moreover, it was shown in [5] that for 1 ≤ p < ∞, any σ-harmonic L p -function associated to an adapted probability measure σ on a locally compact group G is trivial. Harmonic functions on groups play important roles in analysis, geometry and probability theory [6].…”

Contractive projections on Banach algebras

“…For a similar result for locally compact groups, see [7]. The final statement follows from [41,Theorem 6] or [21, Theorem 3, §17].…”

Fixed points and limits of convolution powers of contractive quantum measures

“…Motivated by this result, Chu [Chu] introduced and studied the space of L p -fixed points of the convolution operator L ω for p ∈ [1, ∞). The main result of [Chu] states that if ω is an adapted probability measure, then any such fixed point must be a constant function (see [Chu,Theorem 3.12,Corollary 3.14]).…”