Fractional smoothness of images of logarithmically concave measures under polynomials

TL;DR

This paper demonstrates that images of log-concave measures under degree-d polynomials have densities with fractional smoothness of order 1/d, enabling bounds on total variation distance via the Fortet–Mourier metric.

Contribution

It establishes fractional smoothness properties of polynomial images of log-concave measures and derives related total variation bounds, advancing understanding of measure transformations.

Findings

Measures of polynomial images have densities in Nikol'skii–Besov class of order 1/d.

Total variation distance can be estimated using the Fortet–Mourier distance.

Results apply to real-line measures derived from log-concave measures.

Abstract

We show that a measure on the real line that is the image of a log-concave measure under a polynomial of degree $d$ possesses a density from the Nikol'skii--Besov class of fractional order $1/d$. This result is used to prove an estimate of the total variation distance between such measures in terms of the Fortet--Mourier distance.