Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an embedding between the two?
A decomposition space
D
(
Q
,
L
p
,
Y
)
\mathcal {D}(\mathcal {Q}, L^p, Y)
is determined by a covering
Q
=
(
Q
i
)
i
∈
I
\mathcal {Q} = (Q_i)_{i \in I}
of the frequency domain, an integrability exponent
p
p
, and a sequence space
Y
⊂
C
I
{Y \subset \mathbb {C}^I}
. Given these ingredients, the decomposition space norm of a distribution
g
g
is defined as
‖
g
‖
D
(
Q
,
L
p
,
Y
)
=
‖
(
‖
F
−
1
(
φ
i
⋅
g
^
)
‖
L
p
)
i
∈
I
‖
Y
,
{ \left \Vert g \right \Vert _{\mathcal {D}(\mathcal {Q}, L^p, Y)} = \left \Vert \left ( \left \Vert \mathcal {F}^{-1} \left ( \varphi _{i} \cdot \widehat {g} \right ) \right \Vert _{L^{p}} \right )_{i \in I} \right \Vert _{Y} } ,
where
(
φ
i
)
i
∈
I
(\varphi _i)_{i \in I}
is a suitable partition of unity for
Q
\mathcal {Q}
.
We establish readily verifiable criteria which ensure the existence of a continuous inclusion (“an embedding”)
D
(
Q
,
L
p
1
,
Y
)
↪
D
(
P
,
L
p
2
,
Z
)
\mathcal {D}(\mathcal {Q},L^{p_1},Y) \hookrightarrow \mathcal {D}(\mathcal {P},L^{p_2},Z)
, mostly concentrating on the case where
Y
=
ℓ
w
q
1
(
I
)
Y = \ell _{w}^{q_{1}} ( I )
and
Z
=
ℓ
v
q
2
(
J
)
Z = \ell _{v}^{q_{2}} (J)
. Under suitable assumptions on
Q
,
P
\mathcal {Q}, \mathcal {P}
, we will see that the relevant sufficient conditions are
p
1
≤
p
2
p_{1} \leq p_{2}
and finiteness of a nested norm of the form
\[
‖
(
‖
(
α
i
β
j
⋅
v
j
/
w
i
)
i
∈
I
j
‖
ℓ
t
)
j
∈
J
‖
ℓ
s
,
with
I
j
=
{
i
∈
I
:
Q
i
∩
P
j
≠
∅
}
for
j
∈
J
.
\left \Vert \left ( \left \Vert \left ( \alpha _{i} \beta _{j} \cdot v_{j} / w_{i} \right )_{ i \in I_{j}} \right \Vert _{\ell ^{t}} \right )_{j\in J} \right \Vert _{\ell ^{s}} \, , \quad \text {with} \quad I_{j} = \left \{ i \in I \,:\, Q_{i} \cap P_{j} \neq \varnothing \right \} \quad \text {for } j \in J \,.
\]
Like the sets
I
j
I_j
, the exponents
t
,
s
t, s
and the weights
α
,
β
\alpha , \beta
only depend on the quantities used to define the decomposition spaces.
In a nutshell, in order to apply the embedding results presented in this article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings.
These sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of
p
1
,
p
2
p_1, p_2
, our criteria yield a complete characterization for the existence of the embedding. The same holds for arbitrary values of
p
1
,
p
2
p_1, p_2
under more strict assumptions on the coverings.
We also prove a rigidity result, namely that—for
(
p
1
,
q
1
)
≠
(
2
,
2
)
(p_{1}, q_{1}) \neq (2, 2)
—two decomposition spaces
D
(
Q
,
L
p
1
,
ℓ
w
q
1
)
\mathcal {D}(\mathcal {Q}, L^{p_1}, \ell _w^{q_1})
and
D
(
P
,
L
p
2
,
ℓ
v
q
2
)
\mathcal {D}(\mathcal {P}, L^{p_2}, \ell _v^{q_2})
can only coincide if their “ingredients” are equivalent, that is, if
p
1
=
p
2
p_{1} = p_{2}
and
q
1
=
q
2
q_{1} = q_{2}
and if the coverings
Q
,
P
\mathcal {Q}, \mathcal {P}
and the weights
w
,
v
w, v
are equivalent in a suitable sense.
The resulting embedding theory is illustrated by applications to
α
\alpha
-modulation and Besov spaces. All known embedding results for these spaces are special cases of our approach; often, we improve considerably upon the state of the art.