This article focuses on investigating some basic properties of coupled fractional Fourier transform (CFrFT) on Schwartz-type space. The symbol class \(TS^{m,\alpha ,\beta }_{\mu ,\nu ,\rho ,\sigma }\) is introduced. Continuity of pseudo-differential operators (PDO’s) \(G_{\alpha , \beta }^g\) from \(\mathcal {S}(\mathbb {R}^2)\) onto \(\mathcal {S}_{\alpha ,\beta }(\mathbb {R}^2)\) is given. Integral and kernel representations of \(G_{\alpha , \beta }^g\) are derived. Boundedness property of \(G_{\alpha , \beta }^g\) is discussed on the Sobolev space \(H^{s, \alpha , \beta }(\mathbb {R}^2)\) . The coupled potential operator \(J_s^{\alpha , \beta }\) is defined and it is extended to the space of distributions. Further it is shown that \(J_s^{\alpha , \beta }\) is a continuous mapping from a suitably designed Schwartz-type space into itself. A \(L^p-\) Sobolev like space \(\mathcal {H}_p^{s, \alpha , \beta }(\mathbb {R}^2)\) is introduced and shown as a Banach space. To facilitate the motive, at the end applications of CFrFT to solve Fredholm integral equation and coupled potential in pseudo-differential equation are obtained. We present some examples involving graphs to illustrate the validity of our theoretical findings.
