On twisted Fourier analysis and convergence of Fourier series on discrete groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.Comment: 35 pages; abridged, revised and update

Heat Diffusion on Homogeneous Trees

AbstractLet X be a homogeneous tree. We study the heat diffusion process associated with the nearest neighbour isotropic Markov operator on X. In particular it is shown that the heat maximal operator is weak type (1, 1) and strong type (p, p), for every 1 <p < ∞. We estimate the asymptotic behaviour of the heat maximal function. Moreover, we introduce a family of Hp spaces on X. It is proved that Hp=lp(X) for 1 <p < ∞ and is conjectured that Hp for p less than 1, is trivial

Range of the X-Ray Transform on Trees

AbstractThe X-ray transformation assigns to a function on a space the function induced on the geodesics by integration. We characterize the image of this transformation on trees (not necessarily homogeneous) in terms of algebraic and decay conditions. The case of finite support, or of finite trees, is obtained as a consequence

Universal properties of harmonic functions on trees

We consider an infinite locally finite tree T equipped with nearest neighbor transition coefficients, giving rise to a space of harmonic functions. We show that, except for trivial cases, the generic harmonic function on T has dense range in C. By looking at forward-only transition coefficients, we show that the generic harmonic function induces a boundary martingale that approximates in probability all measurable functions on the boundary of T. We also study algebraic genericity, spaceability and frequent universality of these phenomena. © 2016 Elsevier Inc

Radial heat diffusion from the root of a homogeneous tree and the combinatorics of paths

We compute recursively the heat semigroup in a rooted homogeneous tree for the diffusion with radial (with respect to the root) but non-isotropic transition probabilities. This is the discrete analogue of the heat operator on the disc given by Δ + c ∂/∂r for some constant c that represents a drift towards (or away from) the origin

Frequently dense harmonic functions and universal martingales on trees

On a large class of infinite trees T, we prove the existence of harmonic functions h, with respect to suitable transient transition operators P, that satisfy the following universal property: h is the Poisson transform of a martingale on the end-point boundary Ω of T (equipped with the harmonic measure induced by P) such that, for every measurable function f on Ω, it contains a subsequence converging to f in measure. Moreover, the martingale visits every open set of measurable functions with positive lower density. © 2021 American Mathematical Societ

Twist points of planar domains

We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains. © 2005 American Mathematical Society