Embeddings and compactness for generalized Sobolev–Shubin spaces and modulation spaces

Abstract

For any appropriate weight function ω, spaces are introduced as counterimage of unweighted modulation spaces through Toeplitz operators with symbol ω. It is proved that the weighted modulation spaces coincide with them when ω is a suitable hypoelliptic symbol. It is furhter proved that a necessary and sufficient condition for the embedding

between two modulation spaces to be compact is that the quotient ω₂/ω₁ vanishes at infinity.

Keywords:

• Modulation spaces
• Sobolev
• Shubin
• Compactness
• Toeplitz operators
• Localization operators
• MSC: primary 42B35
• 46E35
• 46B50
• 47B35
• secondary 54D30
• 47L80

Acknowledgements

The first author is thankful for a grant FIRB 2001 and the support of the Department of Mathematics, Blekinge Institute of Technology, Sweden. The second author is thankful for the support of the Department of Mathematics, University of Turin, Italy.