Tauberian theorems for the Stockwell transform of Lizorkin distributions

We prove the continuity of the Stockwell transform and the corresponding synthesis operator on the spaces of highly localized test functions over and , respectively. Using the obtained continuity results, we define and study the Stockwell transform on the space of Lizorkin distributions . We also provide Abelian and Tauberian type results relating the asymptotic behavior of distributions with the asymptotics of their Stockwell transform.