We prove the Kato conjecture for elliptic operators and N×N-systems in divergence form of arbitrary order 2m on \( \Bbb R^n \). More precisely, we assume the coefficients to be bounded measurable and the ellipticity is taken in the sense of a Gårding inequality. We identify the domain of their square roots as the natural Sobolev space \( H^m(\Bbb R^n,\Bbb C^N) \). We also make some remarks on the relation between various ellipticity conditions and Gårding inequality.
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Received May 4, 2001; accepted September 6, 2001.
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Auscher, P., Hofmann, S., McIntosh, A. et al. The Kato square root problem for higher order elliptic operators and systems on $ \Bbb R^n $. J.evol.equ. 1, 361–385 (2001). https://doi.org/10.1007/PL00001377
- Key words: Elliptic systems, Gårding inequality, Kato problem, square roots.