  Derived from the results in (Math. Nachr., 283(12):1758-1770, 2010), in this
paper, we devoted to studying the boundedness properties for Fourier-cosine
convolution weighted by Gaussian functions via Young's type theorem and
Saitoh's type inequality. New norm estimations in the weighted space are
obtained and 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 in $L_1$ space are also found, along with the
analysis of an illustrative example, which exemplifies that the present object
and method solve cases that are not under the conditions of previously known
techniques.
