Weighted symbol spaces, magnetic translations, and pseudodifferential operators

TL;DR

This paper investigates pseudodifferential operators linked to specific symbol spaces with limited regularity, demonstrating their boundedness on modulation spaces and analyzing the Weyl composition product's continuity.

Contribution

It extends the theory of pseudodifferential operators to microlocally defined symbol spaces with limited regularity, providing boundedness results and continuity properties.

Findings

Boundedness of pseudodifferential operators on modulation spaces.

Continuity of the Weyl composition product.

Extension of Sj"ostrand's symbol space framework.

Abstract

We study pseudodifferential operators associated to microlocally defined normed symbol spaces of limited regularity, introduced by J. Sj\"ostrand. Boundedness of such operators on modulation spaces is obtained under suitable conditions, and continuity properties of the Weyl composition product are established.