We study the existence of "L p -type" gradient estimates for the heat kernel of the natural hypoelliptic "Laplacian" on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p > 1. The gradient estimate for p = 2 implies a corresponding Poincaré inequality for the heat kernel. The gradient estimate for p = 1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel.

Abstract. We study heat kernel measures on sub-Riemannian infinitedimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give L p -estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvature-dimension estimate which holds on approximating finite-dimensional projection groups. Such estimates were first introduced by Baudoin and Garofalo in [4].

This paper discusses the existence of gradient estimates for the heat kernel of a second order hypoelliptic operator on a manifold. For elliptic operators, it is now standard that such estimates (satisfying certain conditions on coefficients) are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated "Ricci curvature" takes on the value −∞ at points of degeneracy of the semi-Riemannian metric. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting.This paper presents recent results for hypoelliptic operators. Malliavin calculus methods transfer the problem to one of determining certain infinite dimensional estimates. Here, the underlying manifold is a Lie group, and the hypoelliptic operators are given by the sum of squares of left invariant vector fields. In particular, "L p -type" gradient estimates hold for p ∈ (1, ∞), and the p = 2 gradient estimate implies a Poincaré estimate in this context.

Abstract. Let G denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on G that are square integrable with respect to a heat kernel measure which is formally subelliptic, in the sense that all appropriate finite dimensional projections are smooth measures. We prove a unitary equivalence between a subclass of these square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the "Cameron-Martin" Lie subalgebra. The isomorphism defining the equivalence is given as a composition of restriction and Taylor maps.

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