Upper bound coefficient for convolution structure associated to Hartley--Bessel transform

TL;DR

This paper investigates a convolution structure linked to the Hartley--Bessel transform, establishing bounds and inequalities that improve previous results, and applies these findings to solve certain integral equations.

Contribution

It introduces an improved upper bound coefficient for the convolution $*_{\alpha}$ associated with the Hartley--Bessel transform, extending the theoretical understanding and applications.

Findings

Established an analog of the Hausdorff--Young inequality for the Hartley--Bessel transform.

Proved the convolution $*_{\alpha}$ is uniformly bounded on the dual space.

Provided better upper bound coefficients for the convolution in special cases.

Abstract

This paper is devoted to the study of a convolution structure denoted by $*_{\alpha}$, which is defined via the Hartley--Bessel transform. This concept was introduced in a recent work by F. Bouzeffour [\emph{J. Pseudo-Differ. Oper. Appl.}, 2024;15, Article 42]. We establish an analog of the Hausdorff--Young inequality for the Hartley--Bessel transform and convolution operator $*_{\alpha}$. This leads to the convolution $*_{\alpha}$ being uniformly bounded on the dual space. Moreover, in some special cases, our results yield a better upper bound coefficient for the convolution $*_{\alpha}$ than those previously obtained by Bouzeffour's result in [Theorem 4.4, \emph{J. Pseudo-Differ. Oper. Appl.}, 2024;15, Article 42]. Finally, we apply the convolution structure $*_{\alpha}$ to study the solvability of a particular class of integral equations and provide a priori estimates for solutions under appropriate conditions.