This article focuses on L p estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. We introduce four critical numbers associated to the semigroup and its gradient that completely rule the ranges of exponents for the L p estimates. It appears that the case p < 2 already treated earlier is radically different from the case p > 2 which is new. We thus recover in a unified and coherent way many L p estimates and give further applications. The key tools from harmonic analysis are two criteria for L p boundedness, one for p < 2 and the other for p > 2 but in ranges different from the usual intervals (1, 2) and (2, ∞).

On considère la classe des variétés riemanniennes complètes non compactes dont le noyau de la chaleur satisfait une estimation supérieure et inférieure gaussienne. On montre que la transformée de Riesz y est bornée sur L p , pour un intervalle ouvert de p au-dessus de 2, si et seulement si le gradient du noyau de la chaleur satisfait une certaine estimation L p pour le même intervalle d'exposants p.One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same interval of p's.MSC numbers 2000: 58J35, 42B20