α -Modulation Spaces for Step Two Stratified Lie Groups
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Date
2022Metadata
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- Institutt for matematiske fag [2606]
- Publikasjoner fra CRIStin - NTNU [39956]
Abstract
We define and investigate α-modulation spaces Ms,α p,q(G) associated to a step two stratified Lie group G with rational structure constants. This is an extension of the Euclidean α-modulation spaces Ms,α p,q(Rn) that act as intermediate spaces between the modulation spaces (α = 0) in time-frequency analysis and the Besov spaces (α = 1) in harmonic analysis. We will illustrate that the group structure and dilation structure on G affectthe boundary cases α = 0,1wherethespaces Ms p,q(G)andBs p,q(G)have non-standard translation and dilation symmetries. Moreover, we show that the spaces Ms,α p,q(G) are non-trivial and generally distinct from their Euclidean counterparts. Finally, we examine how the metric geometry of the coverings Q(G) underlying the α =0case Ms p,q(G) allows for the existence of geometric embeddings F : Ms p,q(Rk) −→ Ms p,q(G), as long as k (that only depends on G) is small enough. Our approach naturally gives rise to several open problems that is further elaborated at the end of the paper.