Let \(T_x\) and \(M_\omega\) denote time and frequency shifts, respectively. Let \(V_g(f)\) denote the short-time Fourier transform (STFT) of a function \(f\) with respect to a given window \(g\). We can formally define the Anti-Wick operator (also known under the name of time-frequency localization operator) \(A_a^{\phi_1, \phi_2}\) by the formula: \[ A_a^{\phi_1, \phi_2} (f) = \int_{\mathbb{R}^{2d}} a(x, \omega) V_{\phi_1}(f)(x, \omega) M_\omega T_x (\phi_2) \;d(x ,\omega), \] where \(a\) is called the symbol of the operator \(A_a^{\phi_1, \phi_2}\).
In the paper under review, the authors study conditions under which the above defined operators are bounded operators in \(L^2(\mathbb{R}^d)\), or belong to the Schatten classes. The tools used for the description of such sufficient conditions are the generalized Sobolev spaces (as the spaces for symbols) and modulation spaces (as the spaces for the window functions).