Heat kernel estimates for a class of higher order operators on Lie groups
Let G be a Lie group of polynomial volume growth. Consider a differential operator H of order 2m on G which is a sum of even powers of a generating list A1, . . . , Ad′ of right invariant vector fields. When G is solvable, we obtain an algebraic condition on the list A1, . . . , Ad′ which is sufficient to ensure that the semigroup kernel of H satisfies global Gaussian estimates for all times. For G not necessarily solvable, we state an analytic condition on the list which is necessary and sufficient for global Gaussian estimates. Our results extend previously known results for nilpotent groups.