Boundedness in $L_p$ spaces for the Hartley-Fourier convolutions operator and their applications

Abstract

The paper deals with $L_p$-boundedness of the Hartley-Fourier convolutions operator and their applied aspects. We establish various new Young-type inequalities and obtain the structure of a normed ring in Banach space when equipping it with such convolutional multiplication. Weighted $L_p$-norm inequalities of these convolutions are also considered. As applications, we investigate the solvability and the bounded $L_1$-solution of a class of Fredholm-type integral equations and linear Barbashin's equations with the help of factorization identities of such convolutions. Several examples are provided to illustrate the obtained results to ensure their validity and applicability.