Sharp estimates for trilinear oscillatory integrals and an algorithm of two-dimensional resolution of singularities

Abstract

We obtain sharp estimates for certain trilinear oscillatory integrals, extending Phong and Stein's seminal result to a trilinear setting. This result partially answers a question raised by Christ, Li, Tao and Thiele, concerning sharp estimates for certain multilinear oscillatory integrals. The method in this paper relies on a self-contained algorithm of resolution of singularities in R 2 , which may be of independent interest.

Multilinear Oscillatory Integral Operators and Geometric Stability

“…Likewise (iXϕ −K)T α is equal to T β for some β ∈ A N +1 . This establishes (16). The passage to (14) follows from (15) and (16) by the triangle inequality and the inequality for arithmetic and geometric means:…”

Endpoint estimates for one-dimensional oscillatory integral operators

“…Of course, the ultimate goal is to establish sharp and uniform estimates of (1.4). However our proof of Theorem 1.3 (and thus of Theorem 1.2) relies crucially on the resolution algorithm developed in [35], which itself seems not sufficient to obtain optimal and uniform estimates simultaneously. In particular, the following question remains wide-open: It may be too ambitious to address this problem in such generality.…”

“…Section 3 provides the analytic tools for our main course of the proof. In section 4, we will sketch the proof of the resolution algorithm but refer the readers to [35], as well as [12] for rigorous details. The significance of the algorithm is twofold.…”

“…To get into the heart of our problem, we shall move one step further into the decomposition process, which is often referred as (an algorithm for) resolution of singularities; see Hironaka [13] and many others [4,6,16,23,25,26,34,35]. Our proof of Theorem 1.3 is built up upon the one developed in [35]. The goal of this algorithm is to decompose a small neighborhood of a singularity into finitely many subregions in each of which the function S ′′ xy behaves like a monomial.…”

Higher decay inequalities for multilinear oscillatory integrals

“…One can compare our methods for proving Lemma 2.1 with the more typical resolution of singularities methods of, for example, Greenblatt [7,8], Collins, Greenleaf, and Pramanik [3], or Xiao [18]. Lemma 2.1 allows us to avoid more algebraic considerations by carefully studying the various nonisotropic scalings which make the Taylor polynomials associated to compact faces of the Newton polyhedron homogeneous.…”