Subadditivity of homogeneous norms on certain nilpotent Lie groups

Abstract:

Let $N$ be a Lie group with its Lie algebra generated by the left-invariant vector fields ${X_1}, \ldots ,{X_k}$ on $N$. An explicit fundamental solution for the (hypoelliptic) operator $L = X_1^2 + \cdots + X_k^2$ on $N$ has been obtained for the Heisenberg group by Folland [1] and for the nilpotent (Iwasawa) groups of isometries of rank-one symmetric spaces by Kaplan and Putz [2]. Recently Kaplan [3] introduced a (still larger) class of step-$2$ nilpotent groups $N$ arising from Clifford modules for which similar explicit solutions exist. As in the case of $L$ being the ordinary Laplacian on $N = {{\mathbf {R}}^k}$, these solutions are of the form $g \mapsto {\text {const}}{\left \| g \right \|^{2 - m}}$, $g \in N$, where the "norm" function $\left \| {} \right \|$ satisfies a certain homogeneity condition. We prove that the above norm is also subadditive.