Characterization of smooth symbol classes by Gabor matrix decay

TL;DR

This paper introduces new smooth symbol classes characterized by Gabor matrix decay, linking them to modulation spaces and classical symbol classes, and explores their properties and applications to pseudodifferential and Born-Jordan operators.

Contribution

It characterizes smooth symbol classes via Gabor matrix decay and modulation spaces, extending known classes and providing new boundedness results for related operators.

Findings

Characterization of symbol classes using Gabor matrix decay.

Extension of almost diagonalization properties to new symbol classes.

New boundedness results for Born-Jordan operators.

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 $|\partial^\alpha \sigma(z)|\leq C_\alpha (1+|z|^2)^{m/2}$, $z\in\mathbb{R}^{2d}$, and we show that can be characterized by an intersection of different types of modulation spaces. In the case $m=0$ we recapture the H\"{o}rmander class $S^0_{0,0}$ that can be obtained by intersection of suitable Besov spaces as well. Such spaces contain the Shubin classes $\Gamma^m_\rho$, $0<\rho\leq1$, 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 Gr\"{o}chenig and Rzeszotnik. Finally, we compute the Gabor matrix of a Born-Jordan operator, which allows to prove new boundedness results for such operators.