The Solution of the Kato Square Root Problem for Second Order Elliptic Operators on \Bbb R n

Abstract

We prove the Kato conjecture for elliptic operators on R n. More precisely, we establish that the domain of the square root of a uniformly complex elliptic operator L = −div (A∇) with bounded measurable coefficients in R n is the Sobolev space H 1 (R n) in any dimension with the estimate √ Lf 2 ∼ ∇f 2 .

Quadratic estimates and functional calculi of perturbed Dirac operators

“…This can be proved by abstract methods since −divA∇ is a maximal accretive operator, see [1]. More importantly, Theorem 3.1(ii) implies the full Kato square root estimate [16], and, in full generality, by Auscher-Hofmann-Lacey-M c Intosh-Tchamitchian [2]. Earlier results on the Kato square root problem are due to Fabes-Jerison-Kenig [14] and Coifman-Deng-Meyer [8], where A is assumed to be close to the identity, and to M c Intosh [23] when Hölder continuity of A is assumed.…”

“…By Proposition 4.8, this then suffices to prove Theorems 2.7 and 2.10. This section is an adaptation of the proof of the Kato square root problem for divergence-form elliptic operators [16,2,3], though some estimates require new procedures. For example, we develop new methods based on hypotheses (H5-6) to prove off-diagonal estimates for resolvents of Π B , as the arguments normally used in proving Caccioppoli-type estimates for divergence-form operators do not apply.…”

“…Our result was inspired by the proof of the Kato square root problem by Auscher, Hofmann, Lacey, M c Intosh and Tchamitchian [2], and includes not only it as a corollary, but also many results in the Calderón program such as the boundedness of the Cauchy operator on Lipschitz curves and surfaces. The proof uses many of the concepts developed over the years to prove these results in the Calderón program, and in particular the proof of the Kato problem, but is not a direct consequence, as the operator Π B is first order, and the second order operator Π B 2 is not in divergence form.…”

Remarks on Functional Calculus for Perturbed First-order Dirac Operators

“…The example that motivated the study of perturbed Dirac operators it the following setup, introduced in [7] and exploited in [8] to reprove the Kato square root theorem obtained in [5] for second order operators and in [6] for systems. Let A ∈ L ∞ (R n ; L(C m ⊗ C n )) satisfy …”

Extrapolation of the functional calculus of generalized Dirac operators and related embedding and Littlewood-Paley-type theorems. I

“…The recent complete solution of the long standing square root problem of Kato for elliptic operators and systems by Auscher, Hofmann, Lacey, Mclntosh and Tchamitchian in [7,8,20] was preceded by works of Mclntosh [23] and Coifman, Mclntosh, Meyer [13], as well as a book due to Auscher and Tchamitchian [9] devoted to the boundedness of the square roots of elliptic operators on L 2 .…”