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

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.