Bilinear Sparse Domination for Oscillatory Integral Operators

TL;DR

This paper establishes bilinear sparse domination bounds for a broad class of Fourier integral and oscillatory integral operators, leading to various weighted and weak-type estimates, including for the Schr"odinger operator.

Contribution

It introduces a novel bilinear sparse domination framework applicable to general Fourier and oscillatory integral operators, extending the scope of weighted inequalities.

Findings

Proves sparse bounds for Fourier integral operators of general rank.

Derives weak (1,1) and vector-valued estimates for these operators.

Establishes weighted norm inequalities, including for Schr"odinger operators.

Abstract

In this paper, we prove bilinear sparse domination bounds for a wide class of Fourier integral operators of general rank, as well as oscillatory integral operators associated to H\"ormander symbol classes $S^m_{\rho,\delta}$ for all $0\leq\rho\leq 1$ and $0\leq\delta< 1$, a notable example is the Schr\"odinger operator. As a consequence, one obtains weak $(1,1)$ estimates, vector-valued estimates, and a wide range of weighted norm inequalities for these classes of operators.