Characterization of smooth symbol classes by Gabor matrix decay

Abstract

For $m\in \mathbb{R}$ we introduce the symbol classes $S^m$, $m\in \mathbb{R}$, 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 ${\Gamma }_{\rho }^m$, $0<\rho \le 1$, 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 Gröchenig and Rzeszotnik. Finally, we compute the Gabor matrix of a Born-Jordan operator, which allows to prove new boundedness results for such operators.

Final version, to appear on the Journal of Fourier Analysis and Applications