Oscillation of generalized differences of H\"older and Zygmund functions

In this paper we analyze the oscillation of functions having derivatives in the H\"older or Zygmund class in terms of generalized differences and prove that its growth is governed by a version of the classical Kolmogorov's Law of the Iterated Logarithm. A better behavior is obtained for functions in the Lipschitz class via an interesting connection with Calder\'on-Zygmund operators.Comment: 16 page

On the range of harmonic maps in the plane

In 1994 J. Lewis obtained a purely harmonic proof of the classical Little Picard Theorem by showing that if the joint value distribution of two entire harmonic functions satisfies certain restrictions then they are necessarily constant. We generalize Lewis'theorem and the harmonic Liouville theorem in terms of the range of a harmonic map in the plane

On the Dirichlet problem for solutions of a restricted nonlinear mean value property

Let Ω⊂Rd be a bounded domain and suppose that for each x∈Ω a radius r=r(x) is given so that the ball Bx=B(x,r) is contained in Ω. For 0≤α<1, we consider the following operator in C(¯¯¯¯Ω) Tαu(x)=α2(supBxu+infBxu)+(1−α)∫Bxu, and show that, under certain assumptions on Ω and the radius function r(x), the Dirichlet problem with continuous boundary data has a unique solution u∈C(¯¯¯¯Ω) satisfying Tαu=u. The motivation comes from the study of so called p-harmonious functions and certain stochastic games.Partially supported by grants MTM2011-24606, MTM2014-51824-P and 2014 SGR 75

On the asymptotic mean value property for planar p-harmonic functions

We show that p-harmonic functions in the plane satisfy a nonlinear asymptotic mean value property for p > 1. This extends previous results of Manfredi and Lindqvist for certain range of p’s.Partially supported by grants MTM2011-24606, MTM2014-51824-P and 2014 SGR 75

pp-harmonic functions by way of intrinsic mean value properties

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain satisfying the uniform exterior cone condition. We establish existence and uniqueness of continuous solutions of the Dirichlet Problem associated to certain intrinsic nonlinear mean value properties in $\Omega$. Furthermore we show that, when properly normalized, such functions converge to the $p$-harmonic solution of the Dirichlet problem in $\Omega$, for $p\in[2,\infty)$. The proof of existence is constructive and the methods are entirely analytic, a fundamental tool being the construction of explicit, $p$-independent barrier functions in $\Omega$.Comment: 22 page

A priori Hölder and Lipschitz regularity for generalized p-harmonious functions in metric measure spaces

Let (X, d, μ) be a proper metric measure space and let Ω ⊂ X be a bounded domain. For each x ∈ Ω, we choose a radius 0 < ϱ(x) ≤ dist(x, ∂Ω) and let Bx be the closed ball centered at x with radius ϱ(x). If α ∈ R, consider the following operator in C(Ω), Tαu(x) = α 2 (sup Bx u + inf Bx u) + 1 – α μ(Bx) ∫ Bx u dμ. Under appropriate assumptions on α, X, μ and the radius function ϱ we show that solutions u ∈ C(Ω) of the functional equation Tαu = u satisfy a local Hölder or Lipschitz condition in Ω. The motivation comes from the so called p-harmonious functions in euclidean domains.The research was partially supported by grants MTM2011-24606, MTM2014-51824-p and 2014 SGR 75