Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces

Tom 193 / 2009

Sergei V. Astashkin, Francisco L. Hernández, Evgeni M. Semenov Studia Mathematica 193 (2009), 269-283 MSC: Primary 46E30. DOI: 10.4064/sm193-3-4


If $G$ is the closure of $L_\infty$ in $\exp L_{2}$, it is proved that the inclusion between rearrangement invariant spaces $E\subset F$ is strictly singular if and only if it is disjointly strictly singular and $E\not\supset G$. For any Marcinkiewicz space $M(\varphi) \subset G$ such that $M(\varphi) $ is not an interpolation space between $L_{\infty}$ and $G$ it is proved that there exists another Marcinkiewicz space $M(\psi)\subsetneq M(\varphi)$ with the property that the $M(\psi)$ and $ M(\varphi)$ norms are equivalent on the Rademacher subspace. Applications are given and a question of Milman answered.


  • Sergei V. AstashkinDepartment of Mathematics
    Samara State University
    Samara 443029, Russia
  • Francisco L. HernándezDepartment of Mathematical Analysis
    Madrid Complutense University
    28040 Madrid, Spain
  • Evgeni M. SemenovDepartment of Mathematics
    Voronezh State University
    Voronezh 394006, Russia

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