Large time estimates for heat kernels in nilpotent Lie groups (Q1599939)

Let \(G\) be a connected nilpotent Lie group. Given left invariant vector fields \(X_1,\dots, X_m\) on \(G\) that satisfy the Hörmander condition, the author considers the associated sub-Laplacians \(L= -\sum^m_{i=1} X^2_i+ X_0\), where \(X_0\) is some left invariant vector field. The operator \(L\) determines the semigroup \(e^{-tL}\) whose action is well-known to be given by convolution with a smooth function \(\Phi_t\) known as the heat kernel of \(G\). The purpose of the present paper is to obtain an upper estimate for the heat kernel \(\Phi_t\) for large time (specifically, for all \(t> 1\)). The author establishes a Gaussian upper bound for all such sub-Laplacians \(L\). In particular, the drift term \(X_0\) can be arbitrary. An important tool in the proof is a uniform Harnack inequality for a one-parameter family of sub-Laplacians whose proof is also given in the present paper.