{ "id": "1908.09567", "version": "v1", "published": "2019-08-26T09:45:24.000Z", "updated": "2019-08-26T09:45:24.000Z", "title": "$α$-modulation spaces for step two stratified Lie groups", "authors": [ "Eirik Berge" ], "categories": [ "math.FA", "math.MG" ], "abstract": "We define and investigate $\\alpha$-modulation spaces $M_{p,q}^{s,\\alpha}(G)$ associated to a step two stratified Lie group $G$ with rational structure constants. This is an extension of the Euclidean $\\alpha$-modulation spaces $M_{p,q}^{s,\\alpha}(\\mathbb{R}^n)$ that act as intermediate spaces between the modulation spaces ($\\alpha = 0$) in time-frequency analysis and the Besov spaces ($\\alpha = 1$) in harmonic analysis. We will illustrate that the the group structure and dilation structure on $G$ affect the boundary cases $\\alpha = 0,1$ where the spaces $M_{p,q}^{s}(G)$ and $\\mathcal{B}_{p,q}^{s}(G)$ have non-standard translation and dilation symmetries. Moreover, we show that the spaces $M_{p,q}^{s,\\alpha}(G)$ are non-trivial and generally distinct from their Euclidean counterparts. Finally, we examine how the metric geometry of the coverings $\\mathcal{Q}(G)$ underlying the $\\alpha = 0$ case $M_{p,q}^{s}(G)$ allows for the existence of geometric embeddings \\[F:M_{p,q}^{s}(\\mathbb{R}^k) \\longrightarrow{} M_{p,q}^{s}(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.", "revisions": [ { "version": "v1", "updated": "2019-08-26T09:45:24.000Z" } ], "analyses": { "subjects": [ "46B20", "22E25", "20F65" ], "keywords": [ "stratified lie group", "modulation spaces", "rational structure constants", "open problems", "dilation structure" ], "note": { "typesetting": "TeX", "pages": 0, "language": "en", "license": "arXiv", "status": "editable" } } }