Proceedings of the Centre for Mathematics and its Applications

Stability in $p$ of the $H^\infty$-Calculus of First-Order Systems in $L^p$

Tuomas Hytönen and Alan McIntosh

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Abstract

We study certain differential operators of the form $AD$ arising from a first-order approach to the Kato square root problem. We show that if such operators are$R$-bisectorial in $L^p$, they remain $R-bisectorial in $L^q$ for all $q$ close to $p$.In combination with our earlier results with Portal, which required such $R$-bisectoriality in different $L^q$ spaces to start with, this shows that the $R$-bisectoriality in just one $L^p$ actually implies bounded $H^\infty$-calculus in $L^q$ for all $q$ close to $p$. We adapt the approach to related second-order results developed by Auscher, Hofmann and Martell, and also employ abstract extrapolation theorems due to Kalton and Mitrea

Article information

Source
The AMSI-ANU workshop on spectral theory and harmonic analysis. Andrew Hassell, Alan McIntosh and Robert Taggart, eds. Proceedings of the Centre for Mathematics and its Applications, v. 44. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 2010), 167-181

Dates
First available in Project Euclid: 18 November 2014

Permanent link to this document
https://projecteuclid.org/ euclid.pcma/1416320877

Mathematical Reviews number (MathSciNet)
MR2655384

Zentralblatt MATH identifier
1252.47014

Citation

Hytönen, Tuomas; McIntosh, Alan. Stability in $p$ of the $H^\infty$-Calculus of First-Order Systems in $L^p$. The AMSI–ANU Workshop on Spectral Theory and Harmonic Analysis, 167--181, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 2010. https://projecteuclid.org/euclid.pcma/1416320877


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