The commutators of analysis and interpolation

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