Localization operators and Shapiro-type inequality for the modified Whittaker-Stockwell transform

We define and study the Stockwell transform \(\mathscr{S}_g\) associated with the Whittaker operator \[\Delta_{\alpha}:=-\frac{1}{4}\left[x^2\frac{\mbox{d}^2}{\mbox{d}x^2}+(x^{-1}+(3-4\alpha)x)\frac{\mbox{d}}{\mbox{d}x}\right],\] and prove a Plancherel theorem. Moreover, we define the localization operators \(L_{g,\xi}\) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we give a Shapiro-type uncertainty inequality for the modified Whittaker-Stockwell transform \(\mathscr{S}_g\).