A type of oscillatory integral operator and its applications

TL;DR

This paper investigates $L^p$ estimates for a class of oscillatory integral operators under specific conditions, and applies these results to problems in harmonic analysis and PDEs, including multiplier problems, smoothing estimates, and resolvent bounds.

Contribution

It introduces new $L^p$ estimates for oscillatory integrals satisfying Carleson-Sj"olin conditions with convexity assumptions, and applies these to several key problems in analysis.

Findings

Established $L^p$ bounds for the class of oscillatory operators.

Derived new results for multiplier problems on hypersurfaces.

Obtained sharp resolvent estimates outside the uniform boundedness range.

Abstract

In this paper, we consider $L^p$- estimate for a class of oscillatory integral operators satisfying the Carleson-Sj\"olin conditions with further convex and straight assumptions. As applications, the multiplier problem related to a general class of hypersurfaces with nonvanishing Gaussian curvature, local smoothing estimates for the fractional Schr\"odinger equation and the sharp resolvent estimates outside of the uniform boundedness range are discussed.