This paper extends the study on weighted convolution operators presented in [J. Math. Anal. Appl., vol. 369, no. 2, pp. 712–718, 2010]. Specifically, we focus on the boundedness of a weighted Fourier convolution by one-dimensional Hermite functions via constructing some new \(L_p\) -norm estimates. An extended version of the Young theorem and a Hausdorff–Young type inequality are established. A sharp upper-bound coefficient for such inequalities is computed through Euler Gamma-function. Forward and reverse types of Saitoh inequality for the convolution proposed in this paper over weighted Lebesgue spaces are also formulated. The obtained results for the corresponding convolutions are then utilized to investigate the solvability of Fredholm integral equations and Cauchy-type problems as some applications. Solvability conditions and an explicit solution in \(L_1\) space are formulated. Finally, numerical examples are provided to illustrate the effectiveness of the obtained theoretical results.
