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

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