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The paper deals with $L_p$-boundedness of the Hartley-Fourier convolutionsoperator and their applied aspects. We establish various new Young-typeinequalities and obtain the structure of a normed ring in Banach space whenequipping it with such convolutional multiplication. Weighted $L_p$-norminequalities of these convolutions are also considered. As applications, weinvestigate the solvability and the bounded $L_1$-solution of a class ofFredholm-type integral equations and linear Barbashin's equations with the helpof factorization identities of such convolutions. Several examples are providedto illustrate the obtained results to ensure their validity and applicability.

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