The boundary modulus of continuity of harmonic functions

Modulus of continuity on parts of the boundary and solid modulus of continuity

Majorization of the Modulus of Continuity of Analytic Functions

Let G be a bounded domain in the complex plane, let f be analytic in G and continuous in G, and let μ be a majorant, that is, a non-negative non-decreasing function defined for t ≥ 0 such that μ(2t) ≤ 2μ(t) for all t ≥ 0. Suppose that z1 ∈ ∂G and that |f(z1) − f(z2)| ≤ μ(|z1 − z2|) for all z2 ∈ ∂G. We show that then |f(z1)−f(z2)| ≤ Cμ(|z1−z2|) for all z2 ∈ G where C = 3456. If the assumption is...

The modulus of continuity of analytic functions and CR- geometry

1 Introduction Contour and solid modulus of continuity. The one-dimensional case There has been an extensive study of the relation between the " contour " and the " solid " moduli of continuity for analytic functions in planar domains, starting with the following beautiful result of Hardy and Littlewood. For a domain G in C n (n ≥ 1) we denote as usual by A(G) the algebra of analytic functions ...

Intrinsic Hölder continuity of harmonic functions

In a setting, where only “exit measures” are given, as they are associated with a right continuous strong Markov process on a separable metric space, we provide simple criteria for scaling invariant Hölder continuity of bounded harmonic functions with respect to a distance function which, in applications, may be adapted to the special situation. In particular, already a very weak scaling proper...

Boundary singularities of N -harmonic functions