Sobolev algebras through heat kernel estimates

On a doubling metric measure space (M, d, µ) endowed with a “carré du champ”, let L be the associated Markov generator and L ^p_α(M, L, µ) the corresponding homogeneous Sobolev space of order 0 < α < 1 in L^p, 1 < p < +∞, with norm L ^α/2fp. We give su - cient conditions on the heat semigroup (e⁻t^L )_t>0 for the spaces L ^p_α(M, L, µ) ∩ L^∞(M, µ) to 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^pboundedness of Riesz transforms, but only Lp-boundedness 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.