A variation on the P\'{o}lya-Seg\H{o} principle in one dimension

Abstract

We establish a one-dimensional P\'{o}lya-Szeg\H{o} 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 $\|f'\|_{L^p}$. More precisely, for every measurable function $f$, we prove that its non-increasing rearrangement $f^*$ satisfies $$ \mathcal{V}_p^\alpha(f^*)\le\mathcal{V}_p^\alpha(f) $$ 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.