We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a 3-dimensional Riemannian setting, in either bounded or unbounded domains. With that aim in mind, we introduce a pair of intertwined space-time paraproducts on parabolic Hölder spaces, with good continuity, that happens to be pivotal and provides one of the building blocks of higher order paracontrolled calculus.

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.

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.

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.