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A counterexample in nonlinear interpolation


Author: Michael Cwikel
Journal: Proc. Amer. Math. Soc. 62 (1977), 62-66
MSC: Primary 46M35; Secondary 47H99
DOI: https://doi.org/10.1090/S0002-9939-1977-0433227-0
MathSciNet review: 0433227
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Abstract: If $ ({A_0},{A_1})$ is an interpolation pair with $ {A_0} \subset {A_1}$ and T is a possibly nonlinear operator which maps $ {A_0}$ into $ {A_0}$ and $ {A_1}$ into $ {A_1}$ and satisfies $ {\left\Vert {Ta} \right\Vert _{{A_0}}} \leqslant C{\left\Vert a \right\Vert _{{A_0}}}$ and $ {\left\Vert {Tb - Tb'} \right\Vert _{{A_1}}} \leqslant C{\left\Vert {b - b'} \right\Vert _{{A_1}}}$ for all $ a \in {A_0}$ and b, $ b,b' \in {A_1}$ and for some constant C, then it is known that T also maps the real interpolation spaces $ {({A_0},{A_1})_{\theta ,p}}$ into themselves. We give an example showing that T need not map the complex interpolation spaces $ {[{A_0},{A_1}]_\theta }$ into themselves. It is also seen that quasilinear operators may fail to preserve complex interpolation spaces.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0433227-0
Keywords: Interpolation of nonlinear operators
Article copyright: © Copyright 1977 American Mathematical Society

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