On the absolute continuity of radial projections

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 ${\pi }_{x\mathrm{♯}}\mu$ of $\mu$ are absolutely continuous with respect to ${\mathcal{H}}^{d-1}$ for every centre $x \in $\mathbb{R}$^{d} \setminus \operatorname{spt} \mu$,outsideanexceptionalsetofdimensionatmost$2(d -

1. • s$\mathrm{.}Thisissharp\mathrm{.}Infact,for$x$outsideanexceptionalsetasabove,$\pi_{x\sharp}\mu \in L^{p}(S^{d - 1})$forsome$p > 1$.