Harmonic analysis problems associated with the k-Hankel Gabor transform

We introduce a continuous k-Hankel Gabor transform acting on a Hilbert space deforming L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\mathbb R)$$\end{document}. We prove a Plancherel formula and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-inversion formulas for it. We also prove several uncertainty principles for this transform such as Heisenberg type inequalities and Faris–Price type uncertainty principle.