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

Dorothee Frey; Alan McIntosh; Pierre Portal

Paper

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

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

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

spaces. We also obtain functional calculi results for restrictions to certain subspaces, for a larger range of

. This provides a framework for obtaining

results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator

with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and

bounds on the square-root of

by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in

extends to

for all

, while the restrictions in

come from the operator-theoretic part of the

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.