Sharp $L^p$-Estimates for the Fourier Transform of Surface Measures

\begin{abstract} We consider estimates for the Fourier transform of measures concentrated on smooth surfaces $S\subset \mathbb{R}^3$ given by the graph of a smooth function with simple Arnold singularities such that both principal curvatures of the surface vanish at some point. We prove that if the multiplicity of the critical point of the function does not exceed $7$, then the Fourier transforms of the corresponding surface measures belong to $L^{p}(\mathbb{ R}^3)$ for any $p>3$. Note that for any smooth surface the Fourier transform of a nontrivial surface measure with compact support does not belong to $L^3(\mathbb{R}^3)$; i.e., the $L^p(\mathbb{R}^3)$-estimate obtained is sharp. Moreover, there exists a function with an $E_8$ singularity (the multiplicity of the critical point of the function is equal to $8$) such that the Fourier transform of the corresponding surface measure does not belong to $L^{22/7}(\mathbb{R}^3)$, which shows the sharpness of the estimate for the multiplicity of the critical point.