| Title:
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The commutators of analysis and interpolation (English) |
| Author:
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Cerdà, Joan |
| Language:
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English |
| Journal:
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Nonlinear Analysis, Function Spaces and Applications |
| Volume:
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Vol. 7 |
| Issue:
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2002 |
| Year:
|
|
| Pages:
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21-72 |
| . |
| Category:
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math |
| . |
| Summary:
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The boundedness properties of commutators for operators are of central importance in Mathematical Analysis, and some of these commutators arise in a natural way from interpolation theory. Our aim is to present a general abstract method to prove the boundedness of the commutator $[T,\Omega]$ for linear operators $T$ and certain unbounded operators $\Omega$ that appear in interpolation theory, previously known and a priori unrelated for both real and complex interpolation methods, and also to show how the abstract result applies to some very concrete examples. In Section 1 some examples are given to present some instances where these commutators are used in Analysis. Section 2 is the basic one and contains a general “commutator theorem” for operators of interpolation methods, and the basic idea is that $\Omega$ appears as a combination of two admissible interpolation methods, $\Phi$ and $\Psi$, that correspond to $\Phi(F)=F(\vartheta)$ and $\Psi(f)=F'(\vartheta)$ in the case of the complex method, with $\Omega(f)=\Psi(F)$ if $\Phi(F)=f$ (with a natural boundedness condition over the norms). Section 3 deals with the complex interpolation method and contains typical applications to commutators with pointwise multipliers. Section 4 refers to the real method, and an application to commutators with Fourier multipliers is included. (English) |
| Keyword:
|
Commutator |
| Keyword:
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interpolation |
| Keyword:
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complex method |
| Keyword:
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real method |
| Keyword:
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multiplier |
| MSC:
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42B15 |
| MSC:
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46B70 |
| MSC:
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47B47 |
| . |
| Date available:
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2009-10-08T09:49:37Z |
| Last updated:
|
2012-08-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/702481 |
| . |
| Reference:
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