# Sharp local smoothing estimates for Fourier integral operators

## TL;DR

This paper reviews Fourier integral operators and presents new sharp local smoothing estimates, demonstrating their implications for oscillatory integral estimates and establishing conjectures based on geometric considerations.

## Contribution

The authors establish sharp local smoothing estimates for a class of Fourier integral operators, advancing understanding of their geometric and analytical properties.

## Key findings

- Sharp local smoothing estimates proven for Fourier integral operators
- Implications for oscillatory integral estimates and maximal variants
- Counterexamples confirm the sharpness of the estimates in odd dimensions

## Abstract

The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions arising from the structure of the Fourier integrals.