On trouve des bornes inferieures et superieures au noyau de la chaleur pour une somme de carres de champs vectoriels

We obtain a collection of multilinear Littlewood-Paley estimates, which we then apply to two problems in partial differential equations. The first problem is the estimation of the square root of an elliptic operator in divergence form, and the second is the estimation of solutions to the Cauchy problem for nondivergence-form parabolic equations.

Let L = ∑j = 1mXj2 be sum of squares of vector fields in Rn satisfying a Hörmander condition of order 2: span{Xj, [Xi, Xj]} is the full tangent space at each point. A point x ϵ ∂D of a smooth domain D is characteristic if X1,…, Xm are all tangent to ∂D at x. We prove sharp estimates in non-isotropic Lipschitz classes for the Dirichlet problem near (generic) isolated characteristic points in two special cases: (a) The Grushin operator ∂2∂x2 + x2∂2∂t2 in R2. (b) The real part of the Kohn Laplacian on the Heisenberg group ∑j − 1n (∂∂xj + 2yj∂∂t)2 + (∂∂yj − 2xj∂∂t)2 in R2n + 1. In contrast to non-characteristic points, C∞ regularity may fail at a characteristic point. The precise order of regularity depends on the shape of ∂D at x.