  The purpose of this paper is twofold. The first aim is based on the
Riesz--Thorin interpolation theorem. We establish several of the
Hausdorff--Young type theorems 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.
