Jose Gonzalez: On differentiability of Zygmund and Weierstrass functions

For b > 1 and 0 < a ≤ 1 small enough, Weierstrass proved that the function $$ \sum_{n=0}^{\infty} b^{-an} \cos(b^n x) $$ is nowhere differentiable.
Hardy extended the result to the whole natural range 0 < a ≤ 1, the borderline case a = 1 being the most difficult and interesting. When a =
1 the corresponding Weierstrass function belongs to the so called Zygmund class. The Zygmund class lies in between the Lipschitz spaces $Lip_1$ and $Lip_{\alpha}$ and plays a significant role in a number of problems in Analysis.

Even though Zygmund functions on the real line can be nowhere differentiable, Rachjman and Zygmund remarked that they still have some ”good”differentiability properties in the sense that the divided differences are bounded in many points. Metrically, the optimal result was obtained by Makarov in the 80’s.

In the talk we will consider the analogous problem for Zygmund functions in higher dimensions, with special emphasis to the case of Weierstrass-type functions. (Joint work with Juan J. Donaire and A.
Nicolau, Universitat Autnoma de Barcelona).