Localization operators for the windowed Fourier transform associated with singular partial differential operators

Abstract

We introduce the windowed Fourier transform connected with some singular partial differential operators defined on the half plane $\left [0,+\infty \right [\,\times \mathbb {R}$. Then, we investigate localization operators and show that these operators are not only bounded but also in the Shatten-von Neumann class. We give a trace formula when the symbol function is a nonnegative function.