Remarks on functional calculus for perturbed first order Dirac operators

TL;DR

This paper discusses properties of perturbed first order Dirac operators, focusing on $R$-bisectoriality and functional calculus in $L^p$ spaces, providing a new proof for extrapolation results and insights into space splitting.

Contribution

It offers a new proof of $R$-bisectoriality extrapolation for perturbed Dirac operators and explores its implications on kernel and range splitting.

Findings

New proof of $R$-bisectoriality extrapolation in $L^p$

Impact on space splitting by kernel and range

Enhanced understanding of functional calculus for Dirac operators

Abstract

We make some remarks on earlier works on $R-$bisectoriality in $L^p$ of perturbed first order differential operators by Hyt\"onen, McIntosh and Portal. They have shown that this is equivalent to bounded holomorphic functional calculus in $L^p$ for $p$ in any open interval when suitable hypotheses are made. Hyt\"onen and McIntosh then showed that $R$-bisectoriality in $L^p$ at one value of $p$ can be extrapolated in a neighborhood of $p$. We give a different proof of this extrapolation and observe that the first proof has impact on the splitting of the space by the kernel and range.