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

Abstract

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

32 pages, 1 figure