Damping oscillatory Integrals of convex analytic functions

Let $H\subset \R^{d+1}$ 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 $(\kappa^{1/2} \sigma)^\wedge$ with the damping factor $\kappa^{1/2}$ and prove the optimal decay estimate $$|(\kappa^{1/2} \sigma )^\wedge(\xi)|\le C|\xi|^{-d/2}$$ 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--M\"uller. Furthermore, we prove the same estimates for $(\kappa^{1/2+it} \sigma )^\wedge$ with $C$ growing polynomially in $|t|$. 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 $L^2$--$L^{2(d+2)/(d+4)}$ restriction estimate with respect to the affine surface measure $\kappa^{1/(d+2)} \sigma$. This work was inspired by the stationary set method due to Basu--Guo--Zhang--Zorin-Kranich.