## Abstract

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/}^{2}fp. 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^{p}boundedness 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.

Original language | English |
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Pages (from-to) | 99-161 |

Number of pages | 63 |

Journal | Journal de l'Ecole Polytechnique - Mathematiques |

Volume | 3 |

DOIs | |

Publication status | Published - 2016 |