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Real interpolation of vector-valued spaces in non-diagonal case

Author(s): Irina Asekritova; Natan Krugljak
Journal: Proc. Amer. Math. Soc. 133 (2005), 1665-1675.
MSC (2000): Primary 46B70; Secondary 46E30
Posted: December 20, 2004
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Abstract: It is shown that the formula

\begin{displaymath}(l_{p_{0}}^{s_{0}}(A_{0}),...,l_{p_{n}}^{s_{n}}(A_{n})) _{\vec{\theta },q} =l_{q}^{s}((A_{0},...,A_{n})_{\vec{\theta},q}), \end{displaymath}

where $\vec{\theta}=(\theta _{0},...,\theta _{n})$ and $s=\theta _{0}s_{0}+...+\theta _{n}s_{n}$ is correct under the restrictions $ A_{n-1}=A_{n}$ and $s_{n-1}\neq s_{n}.$ It is also true if we suppose that

\begin{displaymath}A_{n}=(A_{0},A_{1},...,A_{n-1})_{\vec{\lambda},p},s_{n}\neq \lambda _{0}s_{0}+\lambda _{1}s_{1}+...+\lambda _{n-1}s_{n-1}, \end{displaymath}

and the spaces $A_{0},A_{1},...,A_{n-1}$ are functional Banach or quasi-Banach lattices on the same measure space $(\Omega ,\mu ).$

References:

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I. Asekritova and N. Krugljak, On equivalence of $ K$- and $J$-methods for $(n+1)$-tuples of Banach spaces, Studia Math. 122 (1997), 99-116. MR 1432161 (98c:46157)

[AKN]
I. Asekritova, N. Krugljak and L. Nikolova, Lizorkin-Freitag formula for weighted $L_{p}$ spaces and vector-valued interpolation, ( in preparation ).

[AKNMP]
I. Asekritova, N. Krugljak, L. Maligranda, L. Nikolova and L.-E. Persson, Lions-Peetre reiteration formulas for triples and their applications, Studia Math. 145 (2001), 219-254. MR 1829209 (2002e:46089)

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Additional Information:

Irina Asekritova
Affiliation: School of Mathematics and Systems Engineering, Vaxjo University, SE 351 93, Vaxjo, Sweden
Email: irina.asekritova@msi.vxu.se

Natan Krugljak
Affiliation: Department of Mathematics, LuleåUniversity of Technology, SE 972 33, Luleå, Sweden
Email: natan@sm.luth.se

DOI: 10.1090/S0002-9939-04-07714-7
PII: S 0002-9939(04)07714-7
Keywords: Real interpolation, vector-valued spaces
Received by editor(s): September 16, 2003
Posted: December 20, 2004
Communicated by: N. Tomczak-Jaegermann
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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