Notes on restriction theory in the primes

TO BE PUBLISHED BY ISRAEL JOURNAL OF MATHEMATICS. We study the mean $\sum_{x\in\mathcal{X}} \bigl|\sum_{p\le N}{}u_p e(xp)\bigr|^{\ell}$ when $\ell$ covers the full range $[2,\infty)$ and $\mathcal{X}\subset\mathbb{R}/\mathbb{Z}$ is a {well-spaced} set, providing a smooth transition from the case $\ell=2$ to the case $\ell>2$ and improving on the results of J.~Bourgain and of B.~Green and T.~Tao. A uniform Hardy-Littlewood property for the set of primes is established as well as a sharp upper bound for $\sum_{x\in\mathcal{X}} \bigl|\sum_{p\le N}{}u_p e(xp)\bigr|^{\ell}$ when $\mathcal{X}$ is small. These results are extended to primes in \emph{any} interval in a last section, provided the primes are numerous enough therein.