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The Kato square root problem follows from an extrapolation property of the Laplacian

Moritz Egert, Robert Haller-Dintelmann, and Patrick Tolksdorf

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On a domain $\Omega \subseteq \mathbb{R}^d$ we consider second-order elliptic systems in divergence-form with bounded complex coefficients, realized via a sesquilinear form with domain $\mathrm{H}_0^1(\Omega) \subseteq \mathcal{V} \subseteq \mathrm{H}^1(\Omega)$. Under very mild assumptions on~$\Omega$ and $\mathcal{V}$ we show that the solution to the Kato Square Root Problem for such systems can be deduced from a regularity result for the fractional powers of the negative Laplacian in the same geometric setting. This extends earlier results of McIntosh [25] and Axelsson-Keith-McIntosh [6] to non-smooth coefficients and domains.

Article information

Publ. Mat. Volume 60, Number 2 (2016), 451-483.

Received: 4 September 2014
Revised: 2 March 2015
First available in Project Euclid: 11 July 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J57: Boundary value problems for second-order elliptic systems 47A60: Functional calculus 42B37: Harmonic analysis and PDE [See also 35-XX]

Kato's Square Root Problem sectorial and bisectorial operators functional calculus quadratic estimates Carleson measures


Egert, Moritz; Haller-Dintelmann, Robert; Tolksdorf, Patrick. The Kato square root problem follows from an extrapolation property of the Laplacian. Publ. Mat. 60 (2016), no. 2, 451--483. doi:10.5565/PUBLMAT_60216_05.

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