Characterization of smooth symbol classes by Gabor matrix decay. (English) Zbl 1501.47089
Authors’ abstract: For \(m\in\mathbb R\) we consider the symbol classes \(S^m\), consisting of smooth functions \(\sigma\) on \(\mathbb R^{2d}\) such that \(|\partial^{\alpha}\sigma(z)|\le C_{\alpha}(1+|z|^2)^{m/2}\), \(z\in\mathbb R^{2d}\), and we show that can be characterized by an intersection of different types of modulation spaces. In the case \(m=0\) we recapture the Hörmander class \(S^0_{0, 0}\) that can be obtained by intersection of suitable Besov spaces as well. Such spaces contain the Shubin classes \(\Gamma^m_{\rho}\), \(0<\rho\le 1\), and can be viewed as their limit case \(\rho=0\). We exhibit almost diagonalization properties for the Gabor matrix of \(\tau\)-pseudodifferential operators with symbols in such classes, extending the characterization proved by K. Gröchenig and Z. Rzeszotnik [Ann. Inst. Fourier 58, No. 7, 2279–2314 (2008; Zbl 1168.35050)]. Finally, we compute the Gabor matrix of a Born-Jordan operator, which allows to prove new boundedness results for such operators.
Reviewer: Vadim D. Kryakvin (Rostov-na-Donu)
MSC:
| 47G30 | Pseudodifferential operators |
| 42B35 | Function spaces arising in harmonic analysis |
| 81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |
Keywords:
time-frequency analysis; modulation spaces; Gabor matrix; pseudodifferential operators; Gabor framesCitations:
Zbl 1168.35050References:
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