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

Abstract

In this paper, we investigate sharp damping estimates for a class of one dimensional oscillatory integral operators with real-analytic phases. By establishing endpoint estimates for suitably damped oscillatory integral operators, we are able to give a new proof of the sharp L p estimates which have been proved by Xiao in Endpoint estimates for one-dimensional oscillatory integral operators, Advances in Mathematics, 316, 255-291 (2017). The damping estimates obtained in this paper are of independent interest.In this section, we shall first give the concept of horizontally (vertically) convex domains which were introduced by Phong-Stein-Sturm [22]; see also for some related remarks. This notion of convexity turns out to be important in this paper. In fact, the operator van der Corput lemma will be established for oscillatory integral operators supported on horizontally and vertically convex domains. Finally, we shall give a variant of Stein-Weiss interpolation with change of measures. Some related results will be also included in this section.

Sharp $L^p$ decay estimates for degenerate and singular oscillatory integral operators

“…In [14], Shi and Yan established sharp endpoint L p decay for arbitrary homogeneous polynomial phase functions. Later, Xiao extended this result to arbitrary analytic phases in [16], another proof see also [13]. It should be pointed out that our argument relies heavily on the techniques developed in the previous articles.…”

“…For each operator we establish the sharp L 2 decay estimate as well as the endpoint estimate, by adequate interpolations we get the desired results. This idea first appeared in [12] and later was used in [13] to give a new proof of sharp L p decay of real-analytic oscillatory integral operators. The sharp L 2 decay estimates for T z X and T z Y state as follows.…”

Sharp Bounds for Oscillatory Integral Operators with Homogeneous Polynomial Phases

“…This result was extended to the case of smooth phases by Greenblatt [7]; see Rychkov [19] for a partial result. On the other hand, estimates of T λ on L p were also studied by many authors [9,18,28,29,11,23,21,22,5]. Recently, general sharp L p decay estimates have been proved by Xiao [26].…”

A sharp decay estimate for degnerate oscillatory integral operators using broad-narrow method

“…However, the application of this geometric method is limited by the requirement d X " d Y and a frustrated fact is that even the phase function is a simple homogeneous polynomial, the singularities of the corresponding mappings π L and π R are complicated. One way out of the geometric constraints is to focus on the analytic properties, this leads to the thorough understanding of p1 `1q´dimensional operators in the works on L 2 mapping properties [21][22] [23][25] [11] and L p mapping properties [24][31] [27][30] [26].…”