Sobolev algebras through heat kernel estimates

On a doubling metric measure space $(M,d,\mu )$ endowed with a “carr'e du champ”, let $\mathcal{L}$ be the associated Markov generator and $\dot L^{p}\alpha(M,$\mathcal{L}$,\mu)$ the corresponding homogeneous Sobolev space of order $0<\alpha <1$ in $L^p$, $1<p<+\mathrm{\infty }$, with norm ${\parallel {\mathcal{L}}^{\alpha \mathrm{/}2}f\parallel }_p$. We give sufficient conditions on the heat semigroup $(e^{-t\mathcal{L}}{)}_{t>0}$ for the spaces $\dot L^{p}\alpha(M,$\mathcal{L}$,\mu) \cap L^\infty(M,\mu)$ 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 Poincar'e inequalities nor $L^p$-boundedness of Riesz transforms, but only $L^p$-boundedness of the gradient of the semigroup. As a consequence, in the range $p\in (1,2]$, the Sobolev algebra property is shown under Gaussian upper estimates of the heat kernel only.