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

TL;DR

This paper investigates the boundedness of Hartley-Fourier convolutions in $L_p$ spaces, establishing new inequalities, algebraic structures, and applications to integral equations, enhancing understanding of their analytical properties.

Contribution

It introduces new Young-type inequalities, explores the algebraic structure of the convolution operator, and applies these results to solve integral equations.

Findings

Established $L_p$-boundedness of Hartley-Fourier convolutions.

Developed new Young-type inequalities for these convolutions.

Applied results to solve Fredholm-type and Barbashin's equations.

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.