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      Duality for outer \(L^p_\mu(\ell^r)\) spaces and relation to tent spaces

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          Abstract

          We prove that the outer \(L^p_\mu(\ell^r)\) spaces, introduced by Do and Thiele, are isomorphic to Banach spaces, and we show the expected duality properties between them for \(1 < p \leq \infty, 1 \leq r < \infty\) or \(p=r \in \{ 1, \infty \}\) uniformly in the finite setting. In the case \(p=1, 1 < r \leq \infty\), 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 \(L^p_\mu(\ell^r)\). 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 \(\mathbb{R}^d\).

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          Author and article information

          Journal
          16 January 2020
          Article
          2001.05903
          c6ddf1eb-f5ed-4887-b435-af2d0716977a

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          Custom metadata
          42B35, 46E30
          32 pages, 1 figure
          math.CA math.FA

          Functional analysis,Mathematics
          Functional analysis, Mathematics

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