On the absolute continuity of radial projections

TL;DR

This paper proves that for certain measures in high-dimensional spaces, their radial projections are absolutely continuous with respect to surface measure for almost all centers, with a sharp bound on the exceptional set.

Contribution

It establishes sharp conditions under which radial projections of measures are absolutely continuous, extending previous results in geometric measure theory.

Findings

Radial projections are absolutely continuous outside a small exceptional set.

The exceptional set has dimension at most 2(d-1)-s.

Projections belong to L^p spaces for some p > 1.

Abstract

Let $d \geq 2$ and $d - 1 < s < d$. Let $\mu$ be a compactly supported Radon measure in $\mathbb{R}^{d}$ with finite $s$-energy. I prove that the radial projections $\pi_{x\sharp}\mu$ of $\mu$ are absolutely continuous with respect to $\mathcal{H}^{d - 1}$ for every centre $x \in \mathbb{R}^{d} \setminus \operatorname{spt} \mu$, outside an exceptional set of dimension at most $2(d - 1) - s$. This is sharp. In fact, for $x$ outside an exceptional set as above, $\pi_{x\sharp}\mu \in L^{p}(S^{d - 1})$ for some $p > 1$.