# Oscillatory Loomis-Whitney and Projections of Sublevel Sets

## TL;DR

This paper investigates decay estimates for an oscillatory integral operator with a real analytic phase satisfying nondegeneracy conditions, linking these estimates to the volume of sublevel sets and projections in high-dimensional analysis.

## Contribution

It introduces new decay bounds for oscillatory integrals with phases characterized by Newton polyhedra, connecting geometric properties to analytic estimates.

## Key findings

- Maximal decay rates depend on the Newton polyhedron of the phase.
- Volumes of sublevel sets are small relative to projections onto coordinate hyperplanes.
- Results apply to Lebesgue exponents satisfying certain conditions.

## Abstract

We consider an oscillatory integral operator with Loomis-Whitney multilinear form. The phase is real analytic in a neighborhood of the origin in $\mathbb{R}^d$ and satisfies a nondegeneracy condition related to its Newton polyhedron. Maximal decay is obtained for this operator in certain cases, depending on the Newton polyhedron of the phase and the given Lebesgue exponents. Our estimates imply volumes of sublevel sets of such real analytic functions are small relative to the product of areas of projections onto coordinate hyperplanes.