Interpolation of intersections by the real method
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S. V. Astashkin
Translated by: S. V. Kislyakov - St. Petersburg Math. J. 17 (2006), 239-265
- DOI: https://doi.org/10.1090/S1061-0022-06-00902-2
- Published electronically: February 10, 2006
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Abstract:
Let $(X_0,X_1)$ be a Banach couple with $X_0\cap X_1$ dense both in $X_0$ and in $X_1,$ and let $(X_0,X_1)_{\theta ,q}$ $(0<\theta <1,$ $1\le q<\infty )$ denote the real interpolation spaces. Suppose $\psi$ is a linear functional defined on some linear subspace $M\subset X_0+X_1$ and satisfying $\psi \in (X_0\cap X_1)^*,$ $\psi \ne 0.$ Conditions are considered that ensure the natural identity \begin{equation*} (X_0\cap \operatorname {Ker}\psi ,X_1\cap \operatorname {Ker}\psi )_{\theta ,q} =(X_0,X_1)_{\theta ,q}\cap \operatorname {Ker}\psi . \end{equation*} The results obtained provide a solution for the problem posed by N. Krugljak, L. Maligranda, and L.-E. Persson and pertaining to interpolation of couples of intersections that are generated by an integral functional in a weighted $L_p$-scale. Furthermore, an expression is found for the $\mathcal {K}$-functional on a couple of intersections corresponding to a linear functional, and some other related questions are treated.References
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Bibliographic Information
- S. V. Astashkin
- Affiliation: Samara State University, Ul. Akademika Pavlova 1, Samara 443011, Russia
- MR Author ID: 197703
- Email: astashkn@ssu.samara.ru
- Received by editor(s): September 30, 2003
- Published electronically: February 10, 2006
- © Copyright 2006 American Mathematical Society
- Journal: St. Petersburg Math. J. 17 (2006), 239-265
- MSC (2000): Primary 46B70
- DOI: https://doi.org/10.1090/S1061-0022-06-00902-2
- MathSciNet review: 2159583