AbstractWe study the outer $$L^p$$
L
p
spaces introduced by Do and Thiele on sets endowed with a measure and an outer measure. We prove that, in the case of finite sets, for $$1< p \leqslant \infty , 1 \leqslant r < \infty $$
1
<
p
⩽
∞
,
1
⩽
r
<
∞
or $$p=r \in \{ 1, \infty \}$$
p
=
r
∈
{
1
,
∞
}
, the outer $$L^p_\mu (\ell ^r)$$
L
μ
p
(
ℓ
r
)
quasi-norms are equivalent to norms up to multiplicative constants uniformly in the cardinality of the set. This is obtained by showing the expected duality properties between the corresponding outer $$L^p_\mu (\ell ^r)$$
L
μ
p
(
ℓ
r
)
spaces uniformly in the cardinality of the set. Moreover, for $$p=1, 1 < r \leqslant \infty $$
p
=
1
,
1
<
r
⩽
∞
, we exhibit a counterexample to the uniformity in the cardinality of the finite set. We also show that in the upper half space setting the desired properties hold true in the full range $$1 \leqslant p,r \leqslant \infty $$
1
⩽
p
,
r
⩽
∞
. These results are obtained via greedy decompositions of functions in the outer $$L^p_\mu (\ell ^r)$$
L
μ
p
(
ℓ
r
)
spaces. As a consequence, we establish the equivalence between the classical tent spaces $$T^p_r$$
T
r
p
and the outer $$L^p_\mu (\ell ^r)$$
L
μ
p
(
ℓ
r
)
spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on $$\mathbb {R}^d$$
R
d
.
On the Discrete Normal Modes of Quasigeostrophic Theory
