Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in L^p

Abstract: Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in $L^2$ spaces, and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of $L^p$ spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator, to prove that they have a bounded holomorphic functional calculus in those $L^p$ spaces. We also obtain functional calculi results for restrictions to certain subspaces, for a larger range of $p$. This provides a framework for obtaining $L^p$ results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator $L$ with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and $L^p$ bounds on the square-root of $L$ by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in $L^2$ extends to $L^p$ for all $p \in (1,\infty)$, while the restrictions in $p$ come from the operator-theoretic part of the $L^2$ proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces, and about the relationship between conical and vertical square functions.