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.

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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.

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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