Non-differentiability of the convolution of differentiable real functions (Q785251)

The authors provide an example of 2-periodic real differentiable functions \(f,g\) whose convolution \(f*g\) is not differentiable at any point of some perfect set \(P\). This generalizes the recent result from [\textit{P.~Jiménez-Rodríguez} et al., Rev. Mat. Complut. 32, No.~1, 187--193 (2019; Zbl 1457.44003)]. The paper finishes with the following open question: Assume that \(f,g\in L^1[-1,1]\) are differentiable. How small can the set of points of differentiability of \(f*g\) be? Could it be empty? What if we, additionally, assume that \(f'*g\in L^1[-1,1]\)?