An area theorem for a one-dimensional semidirect extension of homogeneous groups

Abstract:

Let $N$ be a homogeneous group [3] and let $\{ {\delta _a}:a \in A = {R^ + }\}$ be the group of dilations of $N$. We prove an area theorem for harmonic functions w.r.t. a class of second-order left-invariant hypoelliptic differential operators $L$ on the semidirect product $S = NA$ with $ax{a^{ - 1}} = {\delta _a}(x),a \in A,x \in N$.