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Lipschitz free $p$-spaces for $0<p<1$ | | Fernando Albiac
; Jose L. Ansorena
; Marek Cuth
; Michal Doucha
; | | Date: |
3 Nov 2018 | | Abstract: | This paper initiates the study of the structure of a new class of $p$-Banach
spaces, $0<p<1$, namely the Lipschitz free $p$-spaces (alternatively called
Arens-Eells $p$-spaces) $mathcal{F}_{p}(mathcal{M})$ over $p$-metric spaces.
We systematically develop the theory and show that some results hold as in the
case of $p=1$, while some new interesting phenomena appear in the case $0<p<1$
which have no analogue in the classical setting. For the former, we, e.g., show
that the Lipschitz free $p$-space over a separable ultrametric space is
isomorphic to $ell_{p}$ for all $0<ple 1$, or that $ell_p$ isomorphically
embeds into $mathcal{F}_p(mathcal{M})$ for any $p$-metric space
$mathcal{M}$. On the other hand, solving a problem by the first author and N.
Kalton, there are metric spaces $mathcal{N}subset mathcal{M}$ such that the
natural embedding from $mathcal{F}_p(mathcal{N})$ to
$mathcal{F}_p(mathcal{M})$ is not an isometry. | | Source: | arXiv, 1811.1265 | | Services: | Forum | Review | PDF | Favorites | |
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