The metaplectic action on modulation spaces

Applied and Computational Harmonic Analysis, vol. 68, pp. 101604

Abstract

We study the mapping properties of metaplectic operators Sˆ∈Mp(2d,R) on modulation spaces of the type Mmp,q(Rd). Our main result is a full characterisation of the pairs (Sˆ,Mp,q(Rd)) for which the operator Sˆ:Mp,q(Rd)→Mp,q(Rd) is (i) well-defined, (ii) bounded. It turns out that these two properties are equivalent, and they entail that Sˆ is a Banach space automorphism. For polynomially bounded weight functions, we provide a simple sufficient criterion to determine whether the well-definedness (boundedness) of Sˆ:Mp,q(Rd)→Mp,q(Rd) transfers to Sˆ:Mmp,q(Rd)→Mmp,q(Rd).

Authors 2

1. Hartmut Führ corresponding Aachen Lehrstuhl für Geometrie und Analysis

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl für Geometrie und Analysis, RWTH Aachen University, D-52056 Aachen, Germany

2. Irina Shafkulovska corresponding

   Austrian Academy of Sciences · University of Vienna · Acoustics Research Institute

   Affiliation as printed

   Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040 Vienna, Austria

   Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria