## 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.)

### Citations:

Zbl 0501.46065; Zbl 0523.42015
Full Text: