Sharp decay estimates for $(2+1)$-dimensional oscillatory integral operators via Newton height

Abstract

We study $(2+1)$-dimensional oscillatory integral operators of the form \[ T_λf(x,y)=\int_{\mathbb{R}}e^{iλP(x,y)t^k}ψ(x,y,t)f(t)dt,\qquad k\geq 1, \] where the phase $P$ is a real-analytic function with a critical point at the origin. We establish the sharp $L^2\to L^2$ decay rate of $\frac{1}{2}\min \{1/h_P,1/k\}$, where $h_P$ denotes Varchenko's Newton height of $P$. The two terms in the minimum reflect a natural competition between the spatial degeneracy of $P$ and the temporal degeneracy of $t^k$; their optimality is confirmed by a Knapp-type and a focusing example, respectively. A $T{T}^{\ast }$ reduction transforms the $L^2$ estimate into a scalar oscillatory integral, allowing Varchenko's theorem to apply directly. Building on this foundation, complex interpolation yields the sharp $L^2\to L^{2k+2}$ bound. Finally, in the regime $h_P\ge k$, we obtain sharp $L^2\to L^p$ decay estimates for all $p$.