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 L² 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 calculus 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 L² extends to L^p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L² 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.

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We consider abstract Sobolev spaces of Bessel-type associated with an operator. In this work, we pursue the study of algebra properties of such functional spaces through the corresponding semigroup. As a follow-up to our previous work, we show that by making use of the property of a 'carré du champ' identity, this algebra property holds in a wider range than previously shown.@en

We consider operators T satisfying a sparse domination property (Formula presented.)with averaging exponents (Formula presented.). We prove weighted strong type boundedness for (Formula presented.) and use new techniques to prove weighted weak type (Formula presented.) boundedness with quantitative mixed (Formula presented.)–(Formula presented.) estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case (Formula presented.) we improve upon their results as we do not make use of a Hörmander condition of the operator T. Moreover, we also establish a dual weak type (Formula presented.) estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.

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We study the boundedness of Riesz transforms in L p for p > 2 on a doubling metric measure space endowed with a gradient operator and an injective, ω-accretive operator L satisfying Davies–Gaffney estimates. If L is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on L p for p in some interval [2, 2 + ), and that L p gradient estimates for the semigroup imply boundedness of the Riesz transform in Lq for q ∈ [2, p). This improves results of Auscher et al. (Ann Sci Ecole Norm Sup 37(4):911–957, 2004) and Auscher and Coulhon (Ann Scuola Norm Sup Pisa 4:531–555, 2005), where the stronger assumption of a Poincaré inequality and the assumption e−t L (1) = 1 were made. The Poincaré inequality assumption is also weakened in the setting of a sectorial operator L. In the last section, we study elliptic perturbations of Riesz transforms.@en

On a doubling metric measure space endowed with a “carré du champ”, we consider LpLp estimates (Gp)(Gp) of the gradient of the heat semigroup and scale-invariant LpLp Poincaré inequalities (Pp)(Pp). We show that the combination of (Gp)(Gp) and (Pp)(Pp) for p≥2p≥2 always implies two-sided Gaussian heat kernel bounds. The case p=2p=2 is a famous theorem of Saloff-Coste, of which we give a shorter proof, without parabolic Moser iteration. We also give a more direct proof of the main result in [37]. This relies in particular on a new notion of LpLp Hölder regularity for a semigroup and on a characterisation of (P2)(P2) in terms of harmonic functions.@en