A variation on the Pólya-Segő principle in one dimension

Sorina Barza; Martin Lind

A variation on the Pólya-Segő principle in one dimension

Classical Analysis and ODEs 2026-07-03 v1

Authors: Sorina Barza , Martin Lind

Abstract

We prove a P\'{o}lya-Szeg\H{o} 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 P\'{o}lya-Szeg\H{o} 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.

Keywords

Pólya-Szegő principle Riesz variation fractional smoothness nowhere differentiable functions Takagi-van der Waerden functions Riemann function

Cite

@article{arxiv.2607.03450,
  title  = {A variation on the Pólya-Segő principle in one dimension},
  author = {Sorina Barza and Martin Lind},
  journal= {arXiv preprint arXiv:2607.03450},
  year   = {2026}
}