On the absolute continuity of radial projections

Abstract

Let $d\ge 2$ 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 ${\pi }_{x♯}\mu$ of $\mu$ are absolutely continuous with respect to ${\mathcal{H}}^{d-1}$ for every centre $x\in {\mathbb{R}}^d\setminus \mathrm{spt}\mu$, 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, ${\pi }_{x♯}\mu \in L^p(S^{d-1})$ for some $p>1$.

6 pages. This paper supersedes arXiv:1602.07629