The P\'olya-Szeg\H{o} inequality for smoothing rearrangements

TL;DR

This paper proves very general versions of the Pólya-Szegő inequality for smoothing rearrangements, encompassing various classes of functions and anisotropic cases, unifying previous results under a broad framework.

Contribution

It introduces comprehensive, general theorems for the Pólya-Szegő inequality applicable to all smoothing rearrangements, unifying and extending prior specific cases.

Findings

Proved general Pólya-Szegő inequalities for smoothing rearrangements.

Covered all main classes of functions previously considered.

Established anisotropic versions involving convex bodies.

Abstract

A basic version of the P\'olya-Szeg\H{o} inequality states that if $\Phi$ is a Young function, the $\Phi$-Dirichlet energy -- the integral of $\Phi(\|\nabla f\|)$ -- of a suitable function $f\in \mathcal{V}(\mathbb{R}^n)$, the class of nonnegative measurable functions on $\mathbb{R}^n$ that vanish at infinity, does not increase under symmetric decreasing rearrangement. This fact, along with variants that apply to polarizations and to Steiner and certain other rearrangements, has numerous applications. Very general versions of the inequality are proved that hold for all smoothing rearrangements, those that do not increase the modulus of continuity of functions. The results cover all the main classes of functions previously considered: Lipschitz functions $f\in \mathcal{V}(\mathbb{R}^n)$, functions $f\in W^{1,p}(\mathbb{R}^n)\cap\mathcal{V}(\mathbb{R}^n)$ (when $1\le p<\infty$ and $\Phi(t)=t^p$), and functions $f\in W^{1,1}_{loc}(\mathbb{R}^n)\cap\mathcal{V}(\mathbb{R}^n)$. In addition, anisotropic versions of these results, in which the role of the unit ball is played by a convex body containing the origin in its interior, are established. Taken together, the results bring together all the basic versions of the P\'olya-Szeg\H{o} inequality previously available under a common and very general framework.