This paper is devoted to the study of solvability and \(L_1\) -estimates for solutions of integral equations of Toeplitz–Hankel and Barbashin types, analyzed through the framework of the polyconvolution structure associated with the Fourier cosine and sine transform domains. In the first part, we establish a new variant of Saitoh’s inequality for this class of polyconvolutions in weighted spaces and compare it with a recent related result in Tuan and Tuan (Integr Transform Spec Funct 34(9):690–702, 2023). In the rest, we derive the necessary and sufficient condition for solvability and explicitly construct the unique \(L_1\) -solution to the aforementioned equations, yielding a priori estimates under the established assumptions. Several illustrative examples are presented to demonstrate the validity and applicability of the obtained results.
