A variation on the Pólya-Segő principle in one dimension

Abstract

We establish a one-dimensional Pólya-Szegő principle for the Riesz $(p,\alpha )$-variation ${\mathcal{V}}_p^{\alpha }$, a family of functionals that interpolates between Wiener $p$-variation and the Sobolev seminorm $\Vert f^{\prime }{\Vert }_{L^p}$. More precisely, for every measurable function $f$, we prove that its non-increasing rearrangement $f^{\ast }$ satisfies $${\mathcal{V}}_p^{\alpha }(f^{\ast })\le {\mathcal{V}}_p^{\alpha }(f)$$ for all $1\le p<\infty$ and $0\le \alpha \le 1-1/p$. This result contains and extends the classical variation-diminishing property of rearrangements from Sobolev spaces to a scale of spaces admitting fractional smoothness and, in particular, applies to functions that are nowhere differentiable. Our methods also yield a sharp rearrangement inequality for a modulus of continuity defined via $p$-variation, providing an analogue of a problem posed by Ul'yanov. Finally, we sketch how our inequality can be useful in the study of nonlinear Fredholm integral equations.

Revised and corrected manuscript