Nathaniel Eldredge

Gradient estimates for the subelliptic heat kernel on H-type groups

We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups G of H-type: |∇Ptf | ≤ KPt(|∇f |) where Pt is the heat semigroup corresponding to the sublaplacian on G, ∇ is the subelliptic gradient, and K is a constant. This extends a result of H.-Q. Li [10] for the Heisenberg group. The proof is based on pointwise heat kernel estimates, and follows an approach used by Bakry, Baudoin, Bonnefont, and Chafäı [3].

Precise estimates for the subelliptic heat kernel on H-type groups

Abstract We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups G of H-type. Specifically, we show that there exist positive constants C 1 , C 2 and a polynomial correction function Q t on G such that C 1 Q t e − d 2 4 t ⩽ p t ⩽ C 2 Q t e − d 2 4 t , where p t is the heat kernel, and d the Carnot–Caratheodory distance on G. We also obtain similar bounds on the norm of its subelliptic gradient | ∇ p t | . Along the way, we record explicit formulas for the distance function d and the subriemannian geodesics of H-type groups.

Widder’s Representation Theorem for Symmetric Local Dirichlet Spaces

In classical PDE theory, Widder’s theorem gives a representation for non-negative solutions of the heat equation on

Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups

\mathbb{R }^n

Left-invariant geometries on SU(2) are uniformly doubling

. We show that an analogous theorem holds for local weak solutions of the canonical “heat equation” on a symmetric local Dirichlet space satisfying a local parabolic Harnack inequality.

On complex H-type Lie algebras

We study the law of a hypoelliptic Brownian motion on an infinite-dimensional Heisenberg group based on an abstract Wiener space. We show that the endpoint distribution, which can be seen as a heat kernel measure, is absolutely continuous with respect to a certain product of Gaussian and Lebesgue measures, that the heat kernel is quasi-invariant under translation by the Cameron-Martin subgroup, and that the Radon-Nikodym derivative is Malliavin smooth.

Hypoelliptic heat kernel inequalities on H-type groups

We show that for a hypoelliptic Dirichlet form operator A on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of A is strongly hypercontractive in the sense of Janson. This extends previous results of Gross to a setting in which the operator A is not holomorphic.

Left-invariant geometries on

Abstract On a stratified Lie group G equipped with hypoelliptic heat kernel measure, we study the behavior of the dilation semigroup on L p spaces of log-subharmonic functions. We consider a notion of strong hypercontractivity and a strong logarithmic Sobolev inequality, and show that these properties are equivalent for any group G . Moreover, if G satisfies a classical logarithmic Sobolev inequality, then both properties hold. This extends similar results obtained by Graczyk, Kemp and Loeb in the Euclidean setting.