What is van der Corput's lemma in higher dimensions?
The authors survey, clarify and place in context a number of multilinear estimates involving a real-valued phase function \(u(x_1,\ldots,x_n)\) of several variables on a domain \(\Omega\). More precisely, the authors are interested in multilinear sublevel estimates of the form \[ \left| \int_{\Omega \cap \{ | u| \leq t \}} f_1(x_1) \dots f_n(x_n)\;dx_1 \dots dx_n\right| \leq C t^\gamma \prod_{i=1}^n \| f_i \| _{L^{p_i}} \] and multilinear van der Corput type estimates of the form \[ \left| \int_\Omega e^{i\lambda u(x)} f_1(x_1) \ldots f_n(x_n)\;dx_1 \ldots dx_n\right| \leq C | \lambda| ^{-\gamma} \prod_{i=1}^n \| f_i \| _{L^{p_i}}. \] Here \(\lambda\) should be thought of as a large real parameter, and \(t\) as a small real parameter, and the objective is to obtain exponents \(\gamma\) which are as good as possible. Ideally we wish the exponent to give a scale-invariant result, but the authors show in most cases that this is not feasible except in a number of simple cases. Related to the goal of scale invariance is the goal of minimizing the number of assumptions on the phase \(u\); typically one only asks for some lower bound on a certain mixed derivative of \(u\), as well as some mild convexity conditions on \(\Omega\) (e.g., that \(\Omega\) is a rectangle). The authors point out that it is also often necessary to enforce an additional complexity bound on \(u\), for instance in controlling how often \(u\) or some derivative of \(u\) can change sign. The authors review a number of existing results in these directions, notably those from [\textit{A. Carbery}, \textit{M. Christ}, and \textit{J. Wright}, J. Am. Math. Soc. 12, No. 4, 981--1015 (1999; Zbl 0938.42008)].
scientific article; zbMATH DE number 1881282