Generalized Anti-Wick operators with symbols in distributional Sobolev spaces. (English) Zbl 1072.47045
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).
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).
Reviewer: Wojciech Czaja (Wien)
MSC:
47G30 | Pseudodifferential operators |
35S05 | Pseudodifferential operators as generalizations of partial differential operators |
46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |