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# Mathematical Research Letters

## Volume 25 (2018)

### Number 3

### Higher decay inequalities for multilinear oscillatory integrals

Pages: 819 – 842

DOI: http://dx.doi.org/10.4310/MRL.2018.v25.n3.a5

#### Authors

#### 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 byCarbery and Wright on any product $\prod^d_{j=1} L^{p_j} (\mathbb{R})$ with each $p_j \geq 2$, extending the known results outside the previously-studied range $\sum^d_{j=1} p^{-1}_j = d-1$. Our theorem assumes a second-order nondegeneracy condition of Varčenko type, and as a corollary reproduces a variant of Varčenko’s theorem and implies Fourier decay estimates for measures of smooth density on degenerate hypersurfaces in $\mathbb{R}^d$.

#### Keywords

Newton polyhedron, multilinear oscillatory integral forms

#### 2010 Mathematics Subject Classification

42B20

The second author is partially supported by NSF grant DMS-1361697 and an Alfred P. Sloan Research Fellowship.

Received 9 December 2016