Duality for outer $L_{\mu }^p({\mathrm{\ell }}^r)$ spaces and relation to tent spaces

We prove that the outer $L_{\mu }^p({\mathrm{\ell }}^r)$ spaces, introduced by Do and Thiele, are isomorphic to Banach spaces, and we show the expected duality properties between them for $1<p\le \mathrm{\infty },1\le r<\mathrm{\infty }$ or $p=r \in { 1, \infty }$uniformlyinthefinitesetting\mathrm{.}Inthecase$p=1, 1 < r \leq \infty$, we exhibit a counterexample to uniformity. We show that in the upper half space setting these properties hold true in the full range $1 \leq p,r \leq \infty$. These results are obtained via greedy decompositions of functions in $L_{\mu }^p({\mathrm{\ell }}^r)$. As a consequence, we establish the equivalence between the classical tent spaces $T_r^p$ and the outer $L_{\mu }^p({\mathrm{\ell }}^r)$ 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 ${\mathbb{R}}^d$.