Fundamental solution for higher‐order homogeneous hypoelliptic operators structured on Hörmander vector fields

We introduce and study a new class of higher‐order differential operators defined on , which are built with Hörmander vector fields, homogeneous w.r.t. a family of dilations (but not left‐invariant w.r.t. any structure of Lie group) and have a structure such that a suitably lifted version of the operator is hypoelliptic. We call these operators “generalized Rockland operators.” We prove that these operators are themselves hypoelliptic and, under a natural condition on the homogeneity degree, possess a global fundamental solution which is jointly homogeneous in and satisfies sharp pointwise estimates. Our theory can be applied also to some higher‐order heat‐type operators and their fundamental solutions.