Stability in p of the H-infinity calculus of first-order systems in L^p

TL;DR

This paper demonstrates that R-bisectoriality of certain differential operators in L^p spaces implies similar properties in neighboring L^q spaces, leading to stability of the H-infinity calculus across these spaces.

Contribution

It establishes that R-bisectoriality in one L^p space ensures bounded H-infinity calculus in L^q for all q near p, extending previous results and employing abstract extrapolation techniques.

Findings

R-bisectoriality in L^p implies R-bisectoriality in L^q for q close to p

Bounded H-infinity calculus extends from L^p to neighboring L^q spaces

Method adapts to second-order operators and uses abstract extrapolation theorems

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-infinity 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.