Damping oscillatory Integrals of convex analytic functions

Let $H\subset {\mathbb{R}}^{d+1}$ be a compact, convex, analytic hypersurface of finite type with a smooth measure $\sigma$ on $H$. Let $\kappa$ denote the Gaussian curvature on $H$. We consider the oscillatory integral $({\kappa }^{1\mathrm{/}2}\sigma {)}^{\wedge }$ with the damping factor ${\kappa }^{1\mathrm{/}2}$ and prove the optimal decay estimate $$\mathrm{\mid }({\kappa }^{1\mathrm{/}2}\sigma {)}^{\wedge }(\xi )\mathrm{\mid }\le C\mathrm{\mid }\xi {\mathrm{\mid }}^{-d\mathrm{/}2}$$ for $d=2,3,$ and with an extra logarithmic factor for $d=4$. Our result provides an essentially complete answer, since such decay estimates generally fail for $d\ge 5$, even for convex analytic hypersurfaces, as shown by Cowling–Disney–Mauceri–Müller. Furthermore, we prove the same estimates for $({\kappa }^{1\mathrm{/}2+it}\sigma {)}^{\wedge }$ with $C$ growing polynomially in $\mathrm{\mid }t\mathrm{\mid }$. As consequences, we obtain the best possible estimates for the convolution, maximal, and adjoint restriction operators associated with $H$, incorporating the mitigating factors of optimal orders. In particular, for $d=2,3$, we prove the $L^2$–$L^{2(d+2)\mathrm{/}(d+4)}$ restriction estimate with respect to the affine surface measure ${\kappa }^{1\mathrm{/}(d+2)}\sigma$. This work was inspired by the stationary set method due to Basu–Guo–Zhang–Zorin-Kranich.