Boundary values of harmonic gradients and differentiability of Zygmund and Weierstrass functions

TL;DR

This paper investigates the differentiability of Zygmund and Weierstrass functions in higher dimensions, showing that the set of points with bounded incremental quotients can have maximal Hausdorff dimension despite potential nowhere differentiability.

Contribution

It establishes conditions under which the set of points with bounded harmonic gradients has maximal Hausdorff dimension for these functions.

Findings

Set of points with bounded incremental quotients can have maximal Hausdorff dimension.

Differentiability properties of Zygmund and Weierstrass functions are characterized in higher dimensions.

Abstract

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.