Lp-Inequalities and Parseval-type relations for the Mehler–Fock transform of general order

Abstract

In this article we study new $L^p$-boundedness properties for the Mehler–Fock transform of general order on the spaces $L^p\left(\right(0,\infty ),e^{\alpha x}dx)$ and $L^p\left(\right(0,\infty ),(1+x{)}^{\gamma }dx)$, $1\le p\le \infty$, and $\alpha ,\gamma \in \mathbb{R}$. We also obtain Parseval-type relations over these spaces.