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

This paper is devoted to the study of a convolution structure denoted by ${\ast }_{\alpha }$, 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 ${\ast }_{\alpha }$. This leads to the convolution ${\ast }_{\alpha }$ being uniformly bounded on the dual space. Moreover, in some special cases, our results yield a better upper bound coefficient for the convolution ${\ast }_{\alpha }$ 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 ${\ast }_{\alpha }$ to study the solvability of a particular class of integral equations and provide a priori estimates for solutions under appropriate conditions.