## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- 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

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

Tuomas Hytönen and Alan McIntosh

#### 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

**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