Sharp Lp decay of oscillatory integral operators with certain homogeneous polynomial phases in several variables

Abstract

We obtain the L p decay of oscillatory integral operators T λ with certain homogeneous polynomial phase of degree d in (n + n)-dimensions. In this paper we require that d > 2n. If d/(d − n) < p < d/n, the decay is sharp and the decay rate is related to the Newton distance. In the case of p = d/n or d/(d − n), we also obtain the almost sharp decay, here "almost" means the decay contains a log(λ) term. For otherwise, the L p decay of T λ is also obtained but not sharp. A counterexample also arises in this paper to show that d/(d − n) ≤ p ≤ d/n is not necessary to guarantee the sharp decay.

Sharp Bounds for Oscillatory Integral Operators with Homogeneous Polynomial Phases

“…In the special case n X = n Y , the damping estimates in the theorem, with D being the Hilbert-Schmidt norm of the Hessian of S, were proved by Xu-Yan [27]. For (1 + 1)−dimensional damping estimates with D = S ′′ xy , we refer the reader to Seeger [20] and Phong-Stein [17].…”

“…Under the rank one condition, we shall extend the L 2 result in Greenleaf-Pramanik-Tang [8] to the L p setting in this paper. In this direction, Xu and Yan [27] considered the special case n X = n Y and obtained sharp L p estimates for T λ .…”

Damping estimates for oscillatory integral operators with real-analytic phases and its applications

“…For linear and multi-linear estimates, we refer the reader to Carbery-Christ-Wright [1], Carbery-Wright [3], Phong-Stein-Sturm [22], Christ-Li-Tao-Thiele [4] and Gressman-Xiao [12]. Some work on higher dimensional oscillatory integral operators can be found in Tang [32], Greenleaf-Pramanik-Tang [9] and Xu-Yan [34]. Other results concerning regularity of Radon transforms associated with S, we refer the reader to Greenleaf-Seeger [11] and Seeger [26,27].…”