Analog version of Hausdorff--Young's theorem for quadratic Fourier transforms and boundedness of oscillatory integral operator

TL;DR

This paper establishes new Hausdorff--Young inequalities for quadratic Fourier transforms and linear canonical transforms, and investigates the boundedness of oscillatory integral operators with polynomial phases.

Contribution

It introduces an analog of Hausdorff--Young's theorem for quadratic Fourier transforms and studies the boundedness of related oscillatory integral operators.

Findings

Proved new Hausdorff--Young type inequalities for quadratic Fourier transforms.

Established boundedness results for oscillatory integral operators with polynomial phases.

Extended classical harmonic analysis results to quadratic and canonical transform settings.

Abstract

The purpose of this paper is twofold. The first aim is based on Riesz--Thorin's interpolation theorem, we prove new Hausdorff--Young type inequalities for the Quadratic Fourier transforms in (Ann. Funct. Anal. 2014;5(1):10--23) and linear canonical transforms in (Mediterr. J. Math. 2018;15,13), which were introduced by Castro et al. The second aim is to investigate the boundedness of the oscillatory integral operator with polynomial phases, which is also presented in the last section of the article.