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We show that harmonic oscillator propagators and fractional Fouriertransforms are essentially the same. We deduce continuity properties and fixtime estimates for such operators on modulation spaces, and apply the resultsto prove Strichartz estimates for the harmonic oscillator propagator whenacting on modulation spaces. Especially we extend some results by Balhara,Cordero, Nicola, Rodino and Thangavelu. We also show that general forms offractional harmonic oscillator propagators are continuous on suitable Pilipovicspaces.

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