A decay estimate for the Fourier transform of certain singular measures in ℝ4 and applications

We consider, for a class of functions φ: ℝ² {0} → ℝ² satisfying a nonisotropic homogeneity condition, the Fourier transform û of the Borel measure on ℝ⁴ defined by

$$\mu \left(E \right) = \int_U {{\chi E}\left({x,\varphi \left(x \right)} \right)} \,dx$$

where E is a Borel set of ℝ⁴ and \(U = \left\{{\left({{t^{{\alpha _1}}},{t^{{\alpha _2}}}s} \right):c < s < d,\,\,0 < t < 1} \right\}\). The aim of this article is to give a decay estimate for û for the case where the set of nonelliptic points of φ is a curve in \(\overline U \backslash \left\{{\bf{0}} \right\}\). From this estimate we obtain a restriction theorem for the usual Fourier transform to the graph of φ∣_U: U → ℝ². We also give L^p-improving properties for the convolution operator T_μf = μ * f.