Decay estimates for weighted oscillatory integrals in R^2

Abstract

In this paper, we study decay estimates for a two-dimensional scalar oscillatory integral with degenerate real-analytic phase and amplitude. Integrals such as these form a model for certain higher-dimensional degenerate oscillatory integrals, for which it is known that many of the two-dimensional results fail. We define an analogue of the Newton distance in the weighted case, and prove that this gives the optimal rate of decay for the weighted oscillatory integral under certain generic hypotheses. When these hypotheses fail, we provide counterexamples to show that the optimal rate of decay may be faster in general. We have obtained bounds for the rate of decay in some of these exceptional cases.

Newton polyhedra and weighted oscillatory integrals with smooth phases

“…In order to reveal a motivation of our investigation, after stating results of Varchenko, we explain some earlier results in [30], [31], [1], [26], [5], [20], which are deeply connected with our study. Note that some conditions in the assumptions of the previous results can be considered as typical cases of the assumptions in our new theorems in Section 4, so they are sometimes more useful for the practical applications.…”

“…Until now, there are not so many studies about the weighted case, but some precise results have been obtained in [31], [1], [26], [5], [24]. In these studies, the Newton polyhedra of both the phase and the weight play important roles.…”

“…In [31], [1], [5], [24] it was made an attempt to generalize the results of Varchenko in [30] as directly as possible in the weighted case under the nondegeneracy condition on the phase. Pramanik and Yang [26] consider the two-dimensional case with the weight of the form g(x) = |h(x)| ǫ , where h is real analytic and ǫ is positive. (This g may not be smooth.)…”

“…In particular, we succeed to give explicit formulae of the coefficient of the leading term of the asymptotic expansion under some appropriate conditions, which reveals that the behavior of oscillatory integrals is decided by some important faces, which are called principal faces, of the Newton polyhedra of the phase and the weight. By the way, Pramanik and Yang [26] also use simultaneous resolution of singularities in their analysis. Indeed, they extend the method of Phong and Stein [25] by considering the expressions of both the phase and the weight in terms of the Puiseux serieses of their roots.…”

Spherical Newton Distance for Oscillatory Integrals With Homogeneous Phase Functions

“…Further results for asymptotic behaviors of oscillatory integrals with large λ were considered in [1]- [7]. Now we introduce polar Newton distance and spherical Newton distance for homogenous polynomials in R 2 and R 3 , respectively.…”