The commutators of analysis and interpolation

Cerdà, Joan

Nonlinear Analysis, Function Spaces and Applications, GDML_Books, (2003), p. 21-72 / Harvested from

Résumé

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.

EUDML-ID : urn:eudml:doc:221847
Mots clés:
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@article{702481,
     title = {The commutators of analysis and interpolation},
     booktitle = {Nonlinear Analysis, Function Spaces and Applications},
     series = {GDML\_Books},
     publisher = {Czech Academy of Sciences, Mathematical Institute},
     address = {Praha},
     year = {2003},
     pages = {21-72},
     url = {http://dml.mathdoc.fr/item/702481}
}

Cerdà, Joan. The commutators of analysis and interpolation, dans Nonlinear Analysis, Function Spaces and Applications, GDML_Books,  (2003), pp. 21-72. http://gdmltest.u-ga.fr/item/702481/

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