Fractional Fourier transforms, harmonic oscillator propagators and Strichartz estimates on Pilipovic and modulation spaces

TL;DR

This paper demonstrates the equivalence of harmonic oscillator propagators and fractional Fourier transforms, establishing their continuity on modulation and Pilipovic spaces, and deriving Strichartz estimates for these operators.

Contribution

It introduces the equivalence between harmonic oscillator propagators and fractional Fourier transforms and extends continuity and Strichartz estimates to modulation and Pilipovic spaces.

Findings

Harmonic oscillator propagators are equivalent to fractional Fourier transforms.

Operators are continuous on modulation and Pilipovic spaces.

Strichartz estimates are established for these operators on modulation spaces.

Abstract

We show that harmonic oscillator propagators and fractional Fourier transforms are essentially the same. We deduce continuity properties and fix time estimates for such operators on modulation spaces, and apply the results to prove Strichartz estimates for the harmonic oscillator propagator when acting on modulation spaces. Especially we extend some results by Balhara, Cordero, Nicola, Rodino and Thangavelu. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable Pilipovic spaces.