Hypoelliptic heat kernel inequalities on the Heisenberg group

@inproceedings{Driver2004HypoellipticHK,
  title={Hypoelliptic heat kernel inequalities on the Heisenberg group},
  author={Bruce K. Driver and Tai Melcher},
  year={2004}
}
We study the existence of “Lp-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. 

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