Fundamental inequalities for the iterated Fourier-cosine convolution with Gaussian weight and its application

Derived from the results in [Giang et al.: \emph{Convolutions for the Fourier transforms with geometric variables and applications}, Math. Nachr. 283(12) (2010), 1758--1770], in this paper, we devoted to studying the boundedness properties for the Fourier-cosine convolution weighted by a Gaussian function of the form $\gamma =\exp(-\frac{1}{2}y^2)$ via Young's type theorem and Saitoh's type inequality. New norm estimations in the weighted space are obtained, and the application of the corresponding class of convolutions in Fredholm's second kind of integral equation is discussed. The conditions for the solvability of this equation on the $L_1$ space are also found, along with the analysis of an illustrative numerical example, which exemplifies that the present object and method solve cases that are not under the conditions of previously known techniques.