Convolution of a measure with itself and a restriction theorem

Let $S_{k}=\left \{ (y,|y|^{k})\colon y \in \mathbf {R}^{n-1} \right \} \subset \mathbf {R}^{n}$ and $\sigma$ be the measure defined by $\langle \sigma , \phi \rangle = \int _{\mathbf {R}^{n-1}}\phi (y, |y|^{k}) dy$. Let $\sigma _{P}$ denote the measure obtained by restricting $\sigma$ to the set $P=[0,\infty )^{n-1}$. We prove estimates on $\sigma _{P}*\sigma _{P}$. As a corollary we obtain results on the restriction to $S_{k} \subset \mathbf {R}^{3}$ of the Fourier transform of functions on $\mathbf {R}^{3}$ for $k\in \mathbf {R}$, $2<k<6$.