Lp + Lq and Lp ∩ Lq are not isomorphic for all 1 ≤ p,q ≤ ∞, p ≠ q

Astashkin, Sergey V.; Maligranda, Lech

doi:10.1016/j.crma.2018.04.019

Functional analysis

L_p + L_q and L_p ∩ L_q are not isomorphic for all 1 ≤ p,q ≤ ∞, p ≠ q

Presented by: Gilles Pisier

Astashkin, Sergey V.¹; Maligranda, Lech ²

¹ Department of Mathematics, Samara National Research University, Moskovskoye shosse 34, 443086, Samara, Russia
² Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden

Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 661-665.

Abstract
Résumé

We prove that if $1 \leq p, q \leq \infty$ , then the spaces $L_{p} + L_{q}$ and $L_{p} \cap L_{q}$ are isomorphic if and only if $p = q$ . In particular, $L_{2} + L_{\infty}$ and $L_{2} \cap L_{\infty}$ are not isomorphic, which is an answer to a question formulated in [2].

Nous prouvons que si $1 \leq p, q \leq \infty$ , alors les espaces $L_{p} + L_{q}$ et $L_{p} \cap L_{q}$ sont isomorphes si et seulement si $p = q$ . En particulier, $L_{2} + L_{\infty}$ et $L_{2} \cap L_{\infty}$ ne sont pas isomorphes, ce qui est une réponse à une question formulée dans [2].

Received: 2018-03-08
Accepted: 2018-04-17
Published online: 2018-04-30

DOI: 10.1016/j.crma.2018.04.019

Author's affiliations:

Astashkin, Sergey V. ¹; Maligranda, Lech ²

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     title = {\protect\emph{L}\protect\textsubscript{\protect\emph{p}} + \protect\emph{L}\protect\textsubscript{\protect\emph{q}} and {\protect\emph{L}\protect\textsubscript{\protect\emph{p}} \ensuremath{\cap} \protect\emph{L}\protect\textsubscript{\protect\emph{q}}} are not isomorphic for all 1 \ensuremath{\leq} \protect\emph{p},\protect\emph{q} \ensuremath{\leq} \ensuremath{\infty}, \protect\emph{p} \ensuremath{\neq} \protect\emph{q}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {661--665},
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     doi = {10.1016/j.crma.2018.04.019},
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Astashkin, Sergey V.; Maligranda, Lech. L_p + L_q and L_p ∩ L_q are not isomorphic for all 1 ≤ p,q ≤ ∞, p ≠ q. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 661-665. doi : 10.1016/j.crma.2018.04.019. https://www.numdam.org/articles/10.1016/j.crma.2018.04.019/

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