José G. Llorente

Regularity Properties of Measures, Entropy and the Law of the Iterated Logarithm

We study regularity properties of a positive measure in euclidean space, such as being absolutely continuous with respect to certain Hausdorff measures, in terms of their dyadic doubling properties. Applications of the main results to the distortion of homeomorphisms of the real line and to the regularity of the harmonic measure for some degenerate elliptic operators are given.

Differentiability of functions in the Zygmund class

Let f be a function in the Zygmund class in the euclidean space. It is proved that the Hausdorff dimension of the set of points where f has bounded divided differences, is bigger or equal to one. Furthermore, if f is in the Small Zygmund class, then the Hausdorff dimension of the set of points where f is differentiable, is bigger or equal to one. The sharpness of these results is also discussed.

Boundary values of harmonic gradients and differentiability of zygmund and weierstrass functions

We study differentiability properties of Zygmund functions and series of Weierstrass type in higher dimensions. While such functions may be nowhere differentiable, we show that, under appropriate assumptions, the set of points where the incremental quotients are bounded has maximal Hausdorff dimension.

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

Abstract 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 B x be the closed ball centered at x with radius ϱ ( x ) . If α ∈ R , consider the following operator in C ( Ω ¯ ) , T α u ( x ) = α 2 sup B x u + inf B x u + 1 − α μ ( B x ) ∫ B x u d μ . Under appropriate assumptions on α , X , μ and the radius function ϱ we show that solutions u ∈ C ( Ω ¯ ) of the functional equation T α u = u satisfy a local Holder or Lipschitz condition in Ω . The motivation comes from the so called p -harmonious functions in euclidean domains.