Let p be a real number greater than one and let G be a connected graph of bounded degree. We introduce the p-harmonic boundary of G and use it to characterize the graphs G for which the constant functions are the only pharmonic functions on G. We show that any continuous function on the pharmonic boundary of G can be extended to a function that is p-harmonic on G. We also give some properties of this boundary that are preserved under rough-isometries. Now let be a finitely generated group. As an application of our results, we characterize the vanishing of the first reduced p -cohomology of in terms of the cardinality of its p-harmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on , the p-harmonic boundary of , and the first reduced p -cohomology of .
Let G be a finitely generated, infinite group, let p > 1, and let Lp(G) denote the Banach space . In this paper we will study the first cohomology group of G with coefficients in Lp(G), and the first reduced Lp-cohomology space of G. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.
Abstract. Let G be a locally compact group. We examine the problem of determining when nonzero functions in L 2 (G) have linearly independent left translations. In particular, we establish some results for the case when G has an irreducible, square integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group and the Weyl-Heisenberg group. We also investigate the case when G has an abelian, closed subgroup of finite index.
Abstract. We formulate a version of the Pompeiu problem in the discrete group setting. Necessary and sufficient conditions are given for a finite collection of finite subsets of a discrete abelian group, whose torsion free rank is less than the cardinal of the continuum, to have the Pompeiu property. We also prove a similar result for nonabelian free groups. A sufficient condition is given that guarantees the harmonicity of a function on a nonabelian free group if it satisfies the mean-value property over two spheres.
Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satis es certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.
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