Holomorphic functional calculus of Hodge-Dirac operators in $Lp$

Abstract

Various PDE estimates, such as maximal regularity or Kato’s square root estimates for elliptic operators, can be obtained by establishing the boundedness of a holomorphic functional calculus for relevant differential operators. The difficulty, when dealing with this functional calculus in an abstract functional analytic setting, is that it is not stable under natural perturbations (such as Linfinity perturbations of the coefficients for a divergence form elliptic operator). Here we show that, if one focus on certain differential operators, stability under such perturbations can be established using harmonic analytic techniques developed during the solution of Kato’s square root problem. We are able to work directly in $L_p$, without relying on extrapolation from $L_2$, thanks to probabilistic ideas. This, and the class of differential operators under consideration, is motivated by potential applications, in particular to boundary value problems. This is joint work with Tuomas Hytonen (Helsinki) and Alan McIntosh (ANU).