Characterization of smooth symbol classes by Gabor matrix decay

Abstract

For m ∈ R we introduce the symbol classes S m , m ∈ R, consisting of smooth functions σ on R 2d such that, 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 Γ m ρ , 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 in [22]. Finally, we compute the Gabor matrix of a Born-Jordan operator, which allows to prove new boundedness results for such operators.