Duality for Outer L^p_mu (ell ^r) Spaces and Relation to Tent Spaces

Abstract

We study the outer 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 or p=r in { 1, infty }, the outer L^p_mu (ell ^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) spaces uniformly in the cardinality of the set. Moreover, for p=1, 1 < r leqslant infty , 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 . These results are obtained via greedy decompositions of functions in the outer L^p_mu (ell ^r) spaces. As a consequence, we establish the equivalence between the classical tent spaces T^p_r and the outer L^p_mu (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.