Characterization of Smooth Symbol Classes by Gabor Matrix Decay

For m∈ R we consider the symbol classes Sm, m∈ R, consisting of smooth functions σ on R2d such that | ∂ασ(z) | ≤ Cα(1 + | z| 2) m/2, z∈ R2d, 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 S0,00 that can be obtained by intersection of suitable Besov spaces as well. Such spaces contain the Shubin classes Γρm, 0 < ρ≤ 1 , and can be viewed as their limit case ρ= 0. We exhibit almost diagonalization properties for the Gabor matrix of τ-pseudodifferential operators with symbols in such classes, extending the characterization proved by Gröchenig and Rzeszotnik (Ann Inst Fourier 58(7):2279–2314, 2008). Finally, we compute the Gabor matrix of a Born–Jordan operator, which allows to prove new boundedness results for such operators.