Restriction of Fractional Derivatives of the Fourier Transform

In this paper, we showed that for suitable $(\beta ,p,s,\mathrm{\ell })$ the $\beta$-order fractional derivative with respect to the last coordinate of the Fourier transform of an $L^p({\mathbb{R}}^n)$ function is in $H^{-s}$ after restricting to a graph of a function with non-vanishing Gaussian curvature provided that the restriction of the Fourier transform of such function to the surface is in $H^{\mathrm{\ell }}$. This is a generalization of the result in \cite{GoldStol}*{Theorem 1.12}.