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

We consider, for a class of functions $φ: \mathbb{R}^{2} \setminus \{ {\bf 0} \} \to \mathbb{R}^{2}$ satisfying a nonisotropic homogeneity condition, the Fourier transform $\hatμ$ 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 $U = \{ (t^{α_1}, t^{α_2}s) : c < s < d, \, 0 < t < 1 \}$. The aim of this article is to give a decay estimate for $\hatμ$, for the case where the set of nonelliptic points of $φ$ 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 $φ_{U} : U \to \mathbb{R}^{2}$. We also give $L^{p}$-improving properties for the convolution operator $T_μ f = μ\ast f$.

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