Higher decay inequalities for multilinear oscillatory integrals

TL;DR

This paper establishes sharp decay inequalities for multilinear oscillatory integrals under nondegeneracy conditions, extending previous results and providing new Fourier decay estimates for measures on degenerate hypersurfaces.

Contribution

It introduces improved decay estimates for multilinear oscillatory integrals under Varchenko-type conditions, broadening the known range of exponents and implications.

Findings

Sharp estimates for multilinear oscillatory integrals with $p_j \,\geq 2$

Extension of results beyond the sum of reciprocals equals $d-1$

Reproduction of Varchenko's theorem and Fourier decay implications

Abstract

In this paper we establish sharp estimates (up to logarithmic losses) for the multilinear oscillatory integral operator studied by Phong, Stein, and Sturm and Carbery and Wright on any product $\prod_{j=1}^d L^{p_j}(\mathbb R)$ with each $p_j \geq 2$, expanding the known results for this operator well outside the previous range $\sum_{j=1}^d p_j^{-1} = d-1$. Our theorem assumes second-order nondegeneracy condition of Varchenko type, and as a corollary reproduces Varchenko's theorem and implies Fourier decay estimates for measures of smooth density on degenerate hypersurfaces in $\mathbb R^d$.