Toeplitz operators associated with the directional short-time Fourier transform and applications

In the present article, we prove a Shapiro uncertainty principle for the directional short-time Fourier transform. Next, we introduce the notion of Toeplitz operators associated with the directional short-time Fourier transform. Particularly, we study the trace class properties of such operators and prove that they belong to the Schatten-von Neumann class. Next, we investigate the boundedness and compactness of these Toeplitz operators in the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{p}$$\end{document}-spaces. Finally, we introduce and study the generalized spectrogram associated with these Toeplitz operators.