This paper investigates the boundedness properties of a convolution operator associated with the fractional Fourier transform (often abbreviated as FrFT) of order \(\alpha \) . The operator under study incorporates chirp weight functions \(\gamma _{1,2}(x)=e^{\pm i(x-a(\alpha )x^{2})}\) and serves as a natural generalization of classical convolution structures within the fractional domain, which was first introduced in Wirel. Pers. Commun. 92, 623–637, (2017). First, we establish a Young-type inequality for this operator, proving that it maps \(L_{p}(\mathbb {R})\times L_{q}(\mathbb {R})\) into \(L_{r}(\mathbb {R})\) for \(p,q,r>1\) satisfying \(1/p+1/q=1+1/r\) , with an explicit constant depending solely on \(\alpha \) . Second, we prove a Hausdorff–Young type inequality which guarantees boundedness into the conjugate space \(L_{s_{1}}(\mathbb {R})\) whenever \(1\le p,q,s\le 2\) and \(1/p_1 +1/q_1=1/s\) , thereby extending the range of exponents to the dual setting. Third, we derive a Saitoh-type weighted inequality valid for all \(p>1\) (including \(p=2\) ), providing \(L_{p}\) -boundedness with respect to suitable weight functions. The proofs rely on the Riesz-Thorin interpolation theorem, Hölder’s inequality, and Plancherel-type identities for the FrFT. These results unify and substantially extend classical convolution estimates to the fractional Fourier framework.
