  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.
