LOCALIZATION OPERATORS AND SHAPIRO’S INEQUALITY FOR THE STURM-LIOUVILLE-STOCKWELL TRANSFORM

We define and study the Stockwell transform \(\mathscr {S}_g\) ${\mathcal{S}}_g$ associated with the Sturm-Liouville operator \(\Delta :=\frac{\text{ d}^2}{\text{ d }x^2}+\frac{A'(x)}{A(x)}\frac{\text{ d }}{\text{ d }x}\) $\mathrm{\Delta }:=\frac{{\text{d}}^2}{\text{d}{x}^2}+\frac{A^{\prime }(x)}{A(x)}\frac{\text{d}}{\text{d}x}$ , where A is a nonnegative function satisfying certain conditions. Moreover, we define the localization operators \(L_{g}(\xi )\) $L_g(\xi )$ associated with 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 Sturm-Liouville-Stockwell transform \(\mathscr {S}_g\) ${\mathcal{S}}_g$ . Some results related to the Hankel-Stockwell transform \(\mathscr {S}_{\alpha ,g}\) ${\mathcal{S}}_{\alpha ,g}$ are deduced.