Abstract
AbstractWe define and investigate $$\alpha $$
α
-modulation spaces $$M_{p,q}^{s,\alpha }(G)$$
M
p
,
q
s
,
α
(
G
)
associated to a step two stratified Lie group G with rational structure constants. This is an extension of the Euclidean $$\alpha $$
α
-modulation spaces $$M_{p,q}^{s,\alpha }({\mathbb {R}}^n)$$
M
p
,
q
s
,
α
(
R
n
)
that act as intermediate spaces between the modulation spaces ($$\alpha = 0$$
α
=
0
) in time-frequency analysis and the Besov spaces ($$\alpha = 1$$
α
=
1
) in harmonic analysis. We will illustrate that the group structure and dilation structure on G affect the boundary cases $$\alpha = 0,1$$
α
=
0
,
1
where the spaces $$M_{p,q}^{s}(G)$$
M
p
,
q
s
(
G
)
and $${\mathcal {B}}_{p,q}^{s}(G)$$
B
p
,
q
s
(
G
)
have non-standard translation and dilation symmetries. Moreover, we show that the spaces $$M_{p,q}^{s,\alpha }(G)$$
M
p
,
q
s
,
α
(
G
)
are non-trivial and generally distinct from their Euclidean counterparts. Finally, we examine how the metric geometry of the coverings $${\mathcal {Q}}(G)$$
Q
(
G
)
underlying the $$\alpha = 0$$
α
=
0
case $$M_{p,q}^{s}(G)$$
M
p
,
q
s
(
G
)
allows for the existence of geometric embeddings $$\begin{aligned} F:M_{p,q}^{s}({\mathbb {R}}^k) \longrightarrow {} M_{p,q}^{s}(G), \end{aligned}$$
F
:
M
p
,
q
s
(
R
k
)
⟶
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.
Funder
NTNU Norwegian University of Science and Technology
Publisher
Springer Science and Business Media LLC
Cited by
1 articles.
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