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Paracommutators and minimal spaces. (English) Zbl 0571.46019

Operators and function theory, Proc. NATO Adv. Study Inst., Lancaster/Engl. 1984, NATO ASI Ser., Ser. C 153, 163-224 (1985).
[For the entire collection see Zbl 0563.00012.]
These lectures summarize some work which the writer has been engaged in the past years (mainly after 1981). Some new results are stated for the first time and numerous open issus (directions for further work) are indicated. Lecture 1 and 2 though contain more or less standard material on interpolation and Besov spaces respectively; they are meant to be read as independent introductions to these topics. Lecture 3 is an account of the new theory of Möbius invariant spaces of holomorphic functions [a program first proposed by J. Arazy and then developed by him and S. Fisher and others, including S. Janson and the writer; see also e.g. the survey of the former two in Lecture Notes Math. 1070, 24-44 (1984; Zbl 0553.46021)]. Especially, there is an application to Hankel operators (forms), a proof (”by handwaving”) of Peller’s well-known trace ideal criterion in the case \(1<p<\infty\), the basic new idea being to exploit the minimality of the Besov space \(B_ 1^{11}({\mathbb{T}})\). The aim of Lecture 4 is to announce the writer’s work with S. Janson generalizing Peller’s theorem to so-called paracommutators. By this we mean a bilinear form (in \(L^ 2({\mathbb{R}}^ n))\) of the form \[ \Gamma (f,g)=(2\pi)^{- n}\iint_{{\mathbb{R}}^ n\times {\mathbb{R}}^ n}\hat b(\xi +\eta)A(\xi,\eta)\hat f(\xi)\hat g(\eta)d\xi d\eta, \] where b is the ”symbol” of \(\Gamma =\Gamma_ b\), A its ”Fourier kernel”. One can prove that \(\Gamma_ b\in S_ p\Leftrightarrow b\in B_ p^{n/p,p}({\mathbb{R}}^ n)\) for \(p>n/N\), where N depends on how fast A ”drops off” on the ”diagonal” \(\{\xi +\eta =0\}\). (A fuller account of this theory is now available as a University of Stockholm pre-print; writer sends copies on request.) Finally, the short Lecture 5 sketches some further considerations suggested by an attempt to extend the previous theory to the case of multilinear forms. (Unfortunately, there are quite a few misprints; the writer will send a (partial?) list of corrections upon request.)

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
46M35 Abstract interpolation of topological vector spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems