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

Sergey V. Astashkin; Lech Maligranda

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

Sergey V. Astashkin ¹; Lech Maligranda 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:

Sergey V. Astashkin ¹; Lech Maligranda 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

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     author = {Sergey V. Astashkin and Lech Maligranda},
     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},
     publisher = {Elsevier},
     volume = {356},
     number = {6},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.019},
     language = {en},
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Sergey V. Astashkin; Lech Maligranda. 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://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.019/

References
Cited by

[1] F. Albiac; N.J. Kalton Topics in Banach Space Theory, Springer-Verlag, New York, 2006

[2] S.V. Astashkin; L. Maligranda $L_{p} + L_{\infty}$ and $L_{p} \cap L_{\infty}$ are not isomorphic for all $1 \leq p < \infty, p \neq 2$ , Proc. Amer. Math. Soc., Volume 146 (2018) no. 5, pp. 2181-2194

[3] C. Bennett; R. Sharpley Interpolation of Operators, Academic Press, Boston, 1988

[4] J. Bergh; J. Löfström Interpolation Spaces. An Introduction, Springer-Verlag, Berlin–New York, 1976

[5] N.L. Carothers; S.J. Dilworth Some Banach space embeddings of classical function spaces, Bull. Aust. Math. Soc., Volume 43 (1991) no. 1, pp. 73-77

[6] S.J. Dilworth Intersection of Lebesgue spaces $L_{1}$ and $L_{2}$ , Proc. Amer. Math. Soc., Volume 103 (1988) no. 4, pp. 1185-1188

[7] S.J. Dilworth A scale of linear spaces related to the $L_{p}$ scale, Ill. J. Math., Volume 34 (1990) no. 1, pp. 140-158

[8] T. Holmstedt Interpolation of quasi-normed spaces, Math. Scand., Volume 26 (1970), pp. 177-199

[9] W.B. Johnson; B. Maurey; G. Schechtman; L. Tzafriri Symmetric structures in Banach spaces, Mem. Amer. Math. Soc., Volume 19 (1979) no. 217 (v+298 p)

[10] M.A. Krasnosel'skiĭ; Ja.B. Rutickiĭ Convex Functions and Orlicz Spaces, Noordhoff, Groningen, The Netherlands, 1961

[11] S.G. Krein; Yu.I. Petunin; E.M. Semenov Interpolation of Linear Operators, Amer. Math. Soc., Providence, RI, 1982

[12] J. Lindenstrauss; L. Tzafriri Classical Banach Spaces, II. Function Spaces, Springer-Verlag, Berlin–New York, 1979

[13] L. Maligranda The K-functional for symmetric spaces, Lect. Notes Math., Volume 1070 (1984), pp. 169-182

[14] L. Maligranda Orlicz Spaces and Interpolation, Semin. Math., vol. 5, University of Campinas, Campinas, Brazil, 1989

[15] M.M. Rao; Z.D. Ren Theory of Orlicz Spaces, Marcel Dekker, New York, 1991

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