Auscher, Pascal; Tchamitchian, Philippe 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) Cited in 4 ReviewsCited in 151 Documents MSC: 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 Keywords:Second order complex elliptic operators; maximal accretive operators; \(H^\infty\)-functional calculus; multilinear expansions; Gaussian estimates; Littlewood-Paley-Stein quadratic functionals; Carleson measures; absolutely bounded mean oscillation; Riesz transforms; Calderón-Zygmund operators; resolvent kernel estimates; square function estimates × Cite Format Result Cite Review PDF