The commutators of analysis and interpolation

Cerdà, Joan

  • Nonlinear Analysis, Function Spaces and Applications, Publisher: Czech Academy of Sciences, Mathematical Institute(Praha), page 21-72

Abstract

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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 , Ω ] for linear operators T and certain unbounded operators Ω 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 Ω appears as a combination of two admissible interpolation methods, Φ and Ψ , that correspond to Φ ( F ) = F ( ϑ ) and Ψ ( f ) = F ' ( ϑ ) in the case of the complex method, with Ω ( f ) = Ψ ( F ) if Φ ( 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.

How to cite

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Cerdà, Joan. "The commutators of analysis and interpolation." Nonlinear Analysis, Function Spaces and Applications. Praha: Czech Academy of Sciences, Mathematical Institute, 2003. 21-72. <http://eudml.org/doc/221847>.

@inProceedings{Cerdà2003,
abstract = {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 $\Psi (f)=F^\{\prime \}(\vartheta )$ 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.},
author = {Cerdà, Joan},
booktitle = {Nonlinear Analysis, Function Spaces and Applications},
keywords = {Commutator; interpolation; complex method; real method; multiplier},
location = {Praha},
pages = {21-72},
publisher = {Czech Academy of Sciences, Mathematical Institute},
title = {The commutators of analysis and interpolation},
url = {http://eudml.org/doc/221847},
year = {2003},
}

TY - CLSWK
AU - Cerdà, Joan
TI - The commutators of analysis and interpolation
T2 - Nonlinear Analysis, Function Spaces and Applications
PY - 2003
CY - Praha
PB - Czech Academy of Sciences, Mathematical Institute
SP - 21
EP - 72
AB - 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 $\Psi (f)=F^{\prime }(\vartheta )$ 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.
KW - Commutator; interpolation; complex method; real method; multiplier
UR - http://eudml.org/doc/221847
ER -

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