Remarks on functional calculus for perturbed first order Dirac operators

Pascal Auscher (1) , Sebastian Stahlhut (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

We make some remarks on earlier works on $R-$bisectoriality in $L^p$ of perturbed first order differential operators by Hytönen, 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. Hytönen 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.

Mots clés: Differential operators with bounded measurable coefficients; extrapolation of norm inequalities; $R$-bisectorial operators; coercivity conditions; kernel/range decomposition

Domaines: Analyse classique [math.CA]; Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

Pascal Auscher, Sebastian Stahlhut. Remarks on functional calculus for perturbed first order Dirac operators. Operator Theory: Advances and Applications, 2014, Operator theory in harmonic and non-commutative analysis, 240, pp.31-43. ⟨hal-00803010⟩