2026/07/17 by Shaozhen Xu
#math.CA
We study (2+1)-dimensional oscillatory integral operators of the form Tλf(x,y)=∫ℝeiλP(x,y)tkψ(x,y,t)f(t)dt, k≥ 1, where the phase P is a real-analytic function with a critical point at the origin. We establish the sharp L2→ L2 decay rate of \frac12min\1/hP, 1/k\, where hP 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 tk; their optimality is confirmed by a Knapp-type and a focusing example, respectively. A TT* reduction transforms the L2 estimate into a scalar oscillatory integral, allowing Varchenko's theorem to apply directly. Building on this foundation, complex interpolation yields the sharp L2→ L2k+2 bound. Finally, in the regime hP≥ k, we obtain sharp L2→ Lp decay estimates for all p.