  We study the regularity of convolution powers for measures supported on Salem
sets, and prove related results on Fourier restriction and Fourier multipliers.
In particular we show that for $\alpha$ of the form ${d}/{n}, n=2,3,\cdots$
there exist $\alpha$-Salem measures for which the $L^2$ Fourier restriction
theorem holds in the range $p\le \frac{2d}{2d-\alpha}$.
  The results rely on ideas of K\"orner. We extend some of his constructions to
obtain upper regular $\alpha$-Salem measures, with sharp regularity results for
$n$-fold convolutions for all $n\in \mathbb N$.
