Rochberg, Richard; Weiss, Guido Derivatives of analytic families of Banach spaces. (English) Zbl 0539.46049 Ann. Math. (2) 118, 315-347 (1983). 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 Cited in 5 ReviewsCited in 44 Documents 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.) Keywords:commutator; weight function; Muckenhoupt class; extension of the classical Riesz-Thorin method; point evaluations; analytic families of Banach spaces; potential operator Citations:Zbl 0501.46065; Zbl 0523.42015 PDFBibTeX XMLCite \textit{R. Rochberg} and \textit{G. Weiss}, Ann. Math. (2) 118, 315--347 (1983; Zbl 0539.46049) Full Text: DOI