Approximate identities on some homogeneous Banach spaces

We study the geometric and approximation properties of Marcinkiewicz spaces and Stepanoff spaces, $1 \leq p < \infty$, as well as others where translations are not isometric but bounded (the bounded $p$-mean spaces) or even unbounded (\Mean0). We construct a function $f$ that belongs to these spaces and has the unusual and unexpected property that all approximate identities $\phi_\varepsilon*f$ converge to $f$ pointwise but they never converge in norm

The Lusin area function and local admissible convergence of harmonic functions on homogeneous trees

We prove admissible convergence to the boundary of functions that are harmonic on a subset of a homogeneous tree by means of a discrete Green formula and an analogue of the Lusin area function

The circle transform on trees

We consider the overdetermined problem of integral geometry on trees given by the transform that integrates functions on a tree over circles, and exhibit difference equations that describe the range. We then show how this problem modifies if we restrict the transform to some natural subcomplex of the complex of circles, proving inversion formulas and characterizing ranges. (C) 2003 Elsevier B.V. All rights reserved

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

Converse mean value theorems on trees and symmetric spaces

Harmonic functions satisfy the mean value property with respect to all integrable radial weights if f is harmonic then hf f h for any such weight h But need a function f that satises this relation with a given nonnegative h b e harmonic By a classical result of Furstenb erg the answer is p ositive for every b ounded f on a Riemannian symmetric space but if the boundedness condition is relaxed then the answer turns out to depend on the weight h In this paper various types of weights are investigated on Euclidean and hyp erb olic spaces as well as on homogeneous and semihomogeneous trees IRf h decays faster than exponentially then the mean value property hf f h does not imply harmonicity of f For weights than exponentially at least a weak converse mean value the eigenfunctions of the Laplace operator which satisfy harmonic The critical case is that of exp onential decay exhibit weights that characterize harmonicity and others that do no

Integral Geometry on Trees

In the hyperbolic disc (or, more generally, in real hyperbolic spaces) we consider the horocyclic Radon transform R and the geodesic Radon transform X. Composition with their respective adjoint operators yields two convolution operators on the disc (with respect to the hyperbolic measure). We describe their convolution kernels in comparison with those of the corresponding operators on a homogeneous tree

A potential theoretic approach to twisting

This paper introduces a geometric, potential theoretic approach to the study of twist point in the boundary of a planar domain. It introduces a map h from a domain D to harmonic functions such that, when z tends to a boundary point w, the limit behaviour of h determines if w is a twist point or is sectorially accessible. The construction is based only on potential-theoretic methods and does not use the Riemann mapping theorem