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

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 $\widehat{\mu }$ of the Borel measure on ${\mathbb{R}}^4$ defined by $$\mu (E)={\int }_U{\chi }_E(x,\phi (x))dx$$ where $E$ is a Borel set of ${\mathbb{R}}^4$ and $U = { (t^{\alpha_1}, t^{\alpha_2}s) : c < s < d, , 0 < t < 1 }$\mathrm{.}Theaimofthisarticleistogiveadecayestimatefor$\hat{\mu}$,forthecasewherethesetofnonellipticpointsof$\varphi$ is a curve in $\bar{U}\setminus \{\mathbf{0}\}$. From this estimate we obtain a restriction theorem for the usual Fourier transform to the graph of ${\phi }_U:U\to {\mathbb{R}}^2$. We also give $L^p$-improving properties for the convolution operator $T_{\mu }f=\mu \ast f$.