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 $\phi :{\mathbb{R}}^2\setminus \{0\}\to {\mathbb{R}}^2$ satisfying a nonisotropic homogeneity condition, the Fourier transform $\hat{\mu }$ of the Borel measure on ${\mathbb{R}}^4$ defined by \[ μ(E) = \int_{U} χ_{E}(x, φ(x)) \, dx \] 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 $\phi$ is a curve in $\bar{U}\setminus \{0\}$. 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 $T_{\mu }f=\mu \ast f$.

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