Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by $\sigma \left(A\right)={\int }_B{\chi }_A(x,\phi \left(x\right))dx$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from $L^p\left(ℝ³\right)$ to $L^q(\Sigma ,d\sigma )$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.

Let $x\left(t\right)=(t,\frac{1}{2}{t}^2,\frac{1}{6}{t}^3)$ a certain curve in ${\mathbf{R}}^3.$ We investigate inequalities of the type $\{\int |\widehat{f}\left(x\right(t\left)\right){|}^{b}dt{\}}^{1/b}\le C\parallel f{\parallel }_a$ for $f\in 𝒮(\mathbf{R}$ 3). Our results improve improve an earlier restriction theorem of Prestini. Various generalizations are also discussed.