A Unified Theory of Commutator Estimates for a Class of Interpolation Methods

Abstract

A general family of interpolation methods is introduced which includes, as special cases, the real and complex methods and also the so-called AE or G 1 and G 2 methods defined by Peetre and Gustavsson-Peetre. Derivation operators O and translation operators R are introduced for all methods of this family. A theorem is proved about

Notes on limits of Sobolev spaces and the continuity of interpolation scales

“…One can also treat in this fashion the interpolation methods introduced in [12]. For the J−method it is also easy to check that [30]).…”

“…[22], [31], [26], and the references therein). For example, in [26] it is established that, with a suitable definition of "distance", for many familiar interpolation functors {F θ } θ∈(0,1) (including the methods of [12]) we have…”

“…It is tempting to attempt a proof of Theorem 2 adapting the proof that (2.30) holds for the interpolation method of Cwikel et al, as given in [26], Theorem 16. However the constants involved in the cancellation lemma of [12] (cf. [12], Lemma 3.11, p. 258) blow up as we approach the boundary.…”

“…However the constants involved in the cancellation lemma of [12] (cf. [12], Lemma 3.11, p. 258) blow up as we approach the boundary. Therefore the constants in [26] could also blow up at the boundary.…”

“…One intriguing question here is if there is a suitable normalization of the Rochberg-Weiss Ω operators (cf. [12], [23], and the references therein) that gives nontrivial commutator results at the end points.…”