# Article

MSC: 42B15, 46B70, 47B47
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Keywords:
Commutator; interpolation; complex method; real method; multiplier
Summary:
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.
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