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- W2969416496 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." @default.
- W2969416496 created "2019-08-29" @default.
- W2969416496 creator A5084714970 @default.
- W2969416496 date "2019-08-26" @default.
- W2969416496 modified "2023-09-27" @default.
- W2969416496 title "$alpha$-modulation spaces for step two stratified Lie groups." @default.
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