  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.
