A decay estimate for the Fourier transform of certain singular measures in $\mathbb{R}^{4}$ and applications

Abstract: We consider, for a class of functions $\varphi : \mathbb{R}^{2} \setminus { {\bf 0} } \to \mathbb{R}^{2}$ satisfying a nonisotropic homogeneity condition, the Fourier transform $\hat{\mu}$ of the Borel measure on $\mathbb{R}^{4}$ defined by [ \mu(E) = \int_{U} \chi_{E}(x, \varphi(x)) \, dx ] where $E$ is a Borel set of $\mathbb{R}^{4}$ and $U = { (t^{\alpha_1}, t^{\alpha_2}s) : c < s < d, \, 0 < t < 1 }$. The aim of this article is to give a decay estimate for $\hat{\mu}$, for the case where the set of nonelliptic points of $\varphi$ is a curve in $\bar{U} \setminus { {\bf 0} }$. From this estimate we obtain a restriction theorem for the usual Fourier transform to the graph of $\varphi_{U} : U \to \mathbb{R}^{2}$. We also give $L^{p}$-improving properties for the convolution operator $T_{\mu} f = \mu \ast f$.