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 μ(E)=∫UχE(x,φ(x))dx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \left(E \right) = \int_U {{\chi E}\left({x,\varphi \left(x \right)} \right)} \,dx$$\end{document} where E is a Borel set of ℝ4 and U={(tα1,tα2s):c<s<d,0<t<1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U = \left\{{\left({{t^{{\alpha _1}}},{t^{{\alpha _2}}}s} \right):c < s < d,\,\,0 < t < 1} \right\}$$\end{document}. 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 U¯\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline U \backslash \left\{{\bf{0}} \right\}$$\end{document}. From this estimate we obtain a restriction theorem for the usual Fourier transform to the graph of φ∣U: U → ℝ2. We also give Lp-improving properties for the convolution operator Tμf = μ * f.