Heat kernels and Riesz transforms on nilpotent Lie groups

We consider pure mth order subcoercive operators with complex coefficients acting on a connected nilpotent Lie group. We derive Gaussian bounds with the correct small time singularity and the optimal large time asymptotic behaviour on the heat kernel and all its derivatives, both right and left. Further we prove that the Riesz transforms of all orders are bounded on the Lp-spaces with p e (1,8). Finally, for second-order operators with real coefficeints we derive matching Gaussian lower bounds and deduce Harnack inequalities valid for all times.