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.
$$\alpha $$-Modulation Spaces for Step Two Stratified Lie Groups