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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Interpolation of intersections by the real method

Author: S. V. Astashkin
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 17 (2005), nomer 2.
Journal: St. Petersburg Math. J. 17 (2006), 239-265
MSC (2000): Primary 46B70
Published electronically: February 10, 2006
MathSciNet review: 2159583
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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

$\displaystyle (X_0\cap \operatorname{Ker}\psi,X_1\cap \operatorname{Ker}\psi)_{\theta,q} =(X_0,X_1)_{\theta,q}\cap \operatorname{Ker}\psi. $

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.

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Additional Information

S. V. Astashkin
Affiliation: Samara State University, Ul. Akademika Pavlova 1, Samara 443011, Russia

Keywords: Interpolation space, real interpolation method, $\mathcal{K}$-functional, dilation indices of a function, spaces of measurable functions, weighted spaces
Received by editor(s): September 30, 2003
Published electronically: February 10, 2006
Article copyright: © Copyright 2006 American Mathematical Society

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