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Identification of some real interpolation spaces

Author: Markus Haase
Journal: Proc. Amer. Math. Soc. 134 (2006), 2349-2358
MSC (2000): Primary 47A60, 47D06
Published electronically: February 17, 2006
MathSciNet review: 2213708
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Abstract: We interrelate the real interpolation spaces associated with the couples $ (X,Y), (X+Y,Y), (X,X\cap Y)$, and $ (X+Y,X\cap Y)$, proving among others the identities

$\displaystyle (X+Y,X)_{\theta,p} \cap (X+Y, Y)_{\theta,p}$ $\displaystyle = (X+Y, X \cap Y)_{\theta,p},$    
$\displaystyle (X+Y,X)_{\theta,p} \cap (X+Y, Y)_{1-\theta,p}$ $\displaystyle = (X,Y)_{\theta,p},$    
$\displaystyle (X,X\cap Y)_{\theta,p} + (Y,X\cap Y)_{\theta,p}$ $\displaystyle = (X+Y, X\cap Y)_{\theta,p},$    
$\displaystyle (X,X\cap Y)_{\theta,p} + (Y,X\cap Y)_{1-\theta,p}$ $\displaystyle = (X,Y)_{\theta,p}$    

for all $ p \in [1,\infty], \theta \in [0,1]$.

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Additional Information

Markus Haase
Affiliation: Abteilung Angewandte Analysis, Universität Ulm, Helmholtzstraße 18, D - 89069 Ulm, Germany
Address at time of publication: Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, I - 56126 Pisa, Italy

Keywords: Interpolation space, K-method, real method of interpolation, intersection property, sectorial operator
Received by editor(s): October 20, 2004
Received by editor(s) in revised form: March 8, 2005
Published electronically: February 17, 2006
Additional Notes: The author gratefully acknowledges the financial support from the EU-Research Training Network “Evolution Equations for Deterministic and Stochastic Systems”, Contract No. HPRN-CT-2002-00281
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.