<p>We provide some Abelian and Tauberian results characterizing the quasiasymptotic behavior of Lizorkin distributions in terms of their Stockwell transform. We prove the continuity of the fractional wavelet transform and the corresponding synthesis operator on the Schwartz spaces and their duals, respectively. Additionally, we establish a connection between fractional Stockwell and fractional wavelet transforms and provide some asymptotic results for the distributional fractional wavelet transform.</p>