We dominate non-integral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototype are Riesz transforms / multipliers or paraproducts associated with a second order elliptic operator. It also applies to such operators whose unweighted continuity is restricted to Lebesgue spaces with certain ranges of exponents (p 0 , q 0 ) where 1 ≤ p 0 < 2 < q 0 ≤ ∞. The norm estimates obtained are powers α of the characteristic used by Auscher and Martell. The critical exponent in this case is p = 1 + p0 q ′ 0 . We prove α = 1 p−p0 when p 0 < p ≤ p and α = q0−1 q0−p when p ≤ p < q 0 . In particular, we are able to obtain the sharp A 2 estimates for non-integral singular operators which do not fit into the class of Calderón-Zygmund operators. These results are new even in the Euclidean space and are the first ones for operators whose kernel does not satisfy any regularity estimate.

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. Aiming that, we introduce a pair of intertwined space-time paraproducts on parabolic Hölder spaces, with good continuity. This constitutes to a first step in building a higher order paracontrolled calculus via semigroup methods.Contents 1 I.Bailleul thanks the U.B.O. for their hospitality, part of this work was written there.

Abstract. On a doubling metric measure space (M, d, µ) endowed with a "carré du champ", let L be the associated Markov generator andL p α (M, L, µ) the corresponding homogeneous Sobolev space of order 0 < α < 1 in L p , 1 < p < +∞, with norm L α/2 f p . We give sufficient conditions on the heat semigroup (e −tL ) t>0to be algebras for the pointwise product. Two approaches are developed, one using paraproducts (relying on extrapolation to prove their boundedness) and a second one through geometrical square functionals (relying on sharp estimates involving oscillations). A chain rule and a paralinearisation result are also given. In comparison with previous results ([29, 11]), the main improvements consist in the fact that we neither require any Poincaré inequalities nor L p -boundedness of Riesz transforms, but only L pboundedness of the gradient of the semigroup. As a consequence, in the range p ∈ (1, 2], the Sobolev algebra property is shown under Gaussian upper estimates of the heat kernel only.

Abstract. On a doubling metric measure space endowed with a "carré du champ", we consider L p estimates (G p ) of the gradient of the heat semigroup and scale-invariant L p Poincaré inequalities (P p ). We show that the combination of (G p ) and (P p ) for p ≥ 2 always implies two-sided Gaussian heat kernel bounds. The case p = 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 L p Hölder regularity for a semigroup and on a characterization of (P 2 ) in terms of harmonic functions.