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 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:Mp,qs(Rk)⟶Mp,qs(G),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} F:M_{p,q}^{s}({\\mathbb {R}}^k) \\longrightarrow {} M_{p,q}^{s}(G), \\end{aligned}$$\\end{document}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.
Highlights
The modulation spaces in time-frequency analysis and theBesov spaces Bsp,q (Rn) in harmonic analysis are invaluable in their own fields
The choice to extend the α-modulation spaces to stratified Lie groups is motivated by the desire to obtain the following two properties for the resulting spaces s,α p,q
It is clear that any stratified Lie group G is nilpotent, that is, the adjoint map adX : g → g given by adX (Y ) := [X, Y ] is a nilpotent linear map for all X ∈ g
Summary
Qα that interpolate between the extreme cases U (Rn) and B(Rn) It is advantageous for several of the applications mentioned above to extend the α-modulation spaces to a setting that include non-uniform translation and dilation symmetries. The choice to extend the α-modulation spaces to stratified Lie groups is motivated by the desire to obtain the following two properties for the resulting spaces. This will allow us to use the Euclidean Fourier transform in the description of the (ii) The fact that any stratified Lie group possesses dilations and a (typically nonabelian) group structure is needed for a satisfying definition of the boundary cases α = 0, 1. Do these coverings reflect some geometric property of the stratified Lie group G in the uniform case α = 0?. Theorem (Main Theorem) Let (Rn, ∗G ) denote a rational stratified Lie group with step less than or equal two.
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