A variation on the Pólya-Segő principle in one dimension

We prove a Pólya-Szegő principle for the Riesz $(p,\alpha )$-variation, a scale of fractional smoothness interpolating between bounded $p$-variation and the Sobolev space $W^{1,p}$. In contrast to the classical Pólya-Szegő inequality, our result also holds for certain nowhere differentiable functions possessing fractional smoothness, including Takagi-van der Waerden-type functions, and Riemann’s ‘‘nondifferentiable’’ function.