Superpolyconvolution operator and applications

Abstract

We introduce a new superpolyconvolution operator for Fourier sine and cosine transforms, characterized by a five-transform factorization property that extends the classical four-transform framework. We establish fundamental properties including $L^1$-boundedness, Young-type inequalities on mixed $L^{p_1}$ spaces, a Parseval identity, and a Titchmarsh-type uniqueness theorem. Applications demonstrate the operator's effectiveness in solving integral and integro-differential equations, including Barbashin-type equations, yielding explicit closed-form solutions.