Sets of $p$-restriction and $p$-spectral synthesis

In this paper we investigate the restriction problem. More precisely, we give sufficient conditions for the failure of a set $E$ in $\mathbb{R}^n$ to have the $p$-restriction property. We also extend the concept of spectral synthesis to $L^p(\mathbb{R}^n)$ for sets of $p$-restriction when $p > 1$. We use our results to show that there are $p$-values for which the unit sphere is a set of $p$-spectral synthesis in $\mathbb{R}^n$ when $n \geq 3$.