Square root problem for divergence operators and related topics. (English) Zbl 0909.35001

Astérisque. 249. Paris: Société Mathématique de France, vii, 172 p. (1998).
Authors’ abstract: We present recent progress on the square root problem of Kato for differential operators in divergence form on \(\mathbb{R}^n\). We discuss topics on functional calculus, heat and resolvent kernel estimates, square function estimates and Carleson measure estimates for square roots. In the first chapter, we show in a quantitative way how the theorems of Aronson-Nash and of De Giorgi are equivalent. In the central chapters, we take advantage of recent development in functional calculus and in harmonic analysis to propose a new point of view on Kato’s problem which allows us to unify previous results and extend them. In the last chapter, we study the associated Riesz transforms, their relation to Caldéron-Zygmund operators and their behavior on \(L^p\)-spaces.
Reviewer: J.Wloka (Kiel)


35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J15 Second-order elliptic equations
47A60 Functional calculus for linear operators
42B25 Maximal functions, Littlewood-Paley theory