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Derivatives of analytic families of Banach spaces. (English) Zbl 0539.46049

The paper develops an extension of the classical Riesz-Thorin method, the new guiding idea being to take point evaluations not only of the function but also of its derivative. Thus in the model case of a linear operator T such that \(\| Tf\|_ p\leq \| f\|_ p\) for \(p=1,\infty\) one obtains the commutator estimate \(\| [T,L]f\|_ 2\leq 2\| f\|_ 2\) where \(Lf=^{def}f\cdot \log | f|.\) The main results are stated for analytic families of Banach spaces [see e.g. R. R. Coifman - M. Cwikel - R. Rochberg - Y. Sagher - G. Weiss, Adv. Math. 43, 203-229 (1982; Zbl 0501.46065)]. A large portion of the paper is devoted to a manifold of ”concrete” applications (illustration); for instance, to commutators [T,b] of a Calderón- Zygmund or a potential operator T with a multiplication operator b (with \(b\in BMO)\) in weighted \(L^ p\) spaces, with weights defined by \(A_ p\)-type conditions; especially, in the latter case one gets a new proof of a result of S. Chanillo’s [Indiana Univ. Math. J. 31, 7-56 (1982; Zbl 0523.42015)].
Reviewer: J.Peetre

MSC:

46M35 Abstract interpolation of topological vector spaces
47B47 Commutators, derivations, elementary operators, etc.
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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