Gaussian bounds for reduced heat kernels of subelliptic operators on nilpotent Lie groups

We obtain Gaussian estimates for the kernels of the semigroups generated by a class of subelliptic operators H acting on Lp(Rk). The class includes anharmonic oscillators and Schrödinger operators with external magnetic fields. The estimates imply an H8-functional calculus for the operator H on Lp with p ¿ ??1,8?? and in many cases the spectral p-independence. Moreover, we show for a subclass of operators satisfying a homogeneity property that the Riesz transforms of all orders are bounded.