Lower bounds for parameter-dependent oscillatory integrals and Fourier frames on hypersurfaces

We prove lower bounds for Fourier transforms of surface measures $$\mathrm{\mid }\widehat{\psi d\sigma }(\xi )\mathrm{\mid }{\gtrsim }_{\psi ,K,\sigma ,R_0}\sigma (B(p,\mathrm{\mid }\xi {\mathrm{\mid }}^{-1})),\text{ given }p\in K⋐\{\psi >0\},{\vec{n}}_p\parallel \xi ,\mathrm{\mid }\xi \mathrm{\mid }\ge R_0,$$ on convex hypersurfaces of finite type with at most one vanishing principal curvature at each point. These bounds are uniform in the choice of $p\in K⋐\{\psi >0\}$ and match the classical upper bound of Bruna-Nagel-Wainger. The key new analytic ingredient is a uniform lower bound for parameter-dependent oscillatory integrals on $\mathbb{R}$ with convex phases. We also give convex examples of finite type with two vanishing principal curvatures for which $$\underset{p\in K}{\inf }\mathrm{\mid }\widehat{\psi d\sigma }(R{\vec{n}}_p)\mathrm{\mid }{\lesssim }_N{R}^{-N},\text{ for arbitrary fixed }\{0\}⋐K⋐\{\psi >0\}\mathrm{.}$$ Though the estimate above has its own interest, our original motivation is to study Fourier frames for surface measures. In particular, we show that, if $\mathrm{\Gamma }$ is a closed convex smooth surface in ${\mathbb{R}}^d,d\ge 2$, of finite type and at most one principal curvature vanishes at each point, then the surface measure on $\mathrm{\Gamma }$ does not admit any Fourier frame. This weakens the non-vanishing curvature condition in a previous result of Iosevich, Lai, the second author, and Wyman. On centrally symmetric surfaces of non-vanishing Gaussian curvature, we obtain more domains without Fourier frames. For example, we show that $\{x\in S^{d-1}:0<x_d<\delta \},d\ge 3$, does not admit any Fourier frame for all $\delta >0$, which generalizes the result of Chen and the second author on the hemisphere.