Pseudodifferential operators with symbols in the H\"ormander class $S^0_{\alpha,\alpha}$ on $\alpha$-modulation spaces

TL;DR

This paper investigates the boundedness of pseudodifferential operators with symbols in the H"ormander class $S^0_{ ho, ho}$ on $eta$-modulation spaces, establishing boundedness results for symbols in $S^0_{eta,eta}$ across all $eta$-modulation spaces.

Contribution

It demonstrates that pseudodifferential operators with symbols in $S^0_{eta,eta}$ are bounded on all $eta$-modulation spaces $M_{p,q}^{s,eta}$ for any real $s$ and all exponents $p,q$.

Findings

Operators with symbols in $S^0_{eta,eta}$ are bounded on all $M_{p,q}^{s,eta}$ spaces.

Boundedness holds for arbitrary $s o ext{all real numbers}$.

Results clarify the relation between $eta$ and $ ho$ in symbol classes.

Abstract

In this paper, we study the boundedness of pseudodifferential operators with symbols in the H\"ormander class $S^0_{\rho,\rho}$ on $\alpha$-modulation spaces $M_{p,q}^{s,\alpha}$, and consider the relation between $\alpha$ and $\rho$. In particular, we show that pseudodifferential operators with symbols in $S^0_{\alpha,\alpha}$ are bounded on all $\alpha$-modulation spaces $M^{s,\alpha}_{p,q}$, for arbitrary $s\in\mathbb{R}$ and for the whole range of exponents $0 < p,q \leq \infty$.