Inner functions on the spaces of homogeneus type

In the article the M. Hakim–N. Sibony–B. Low construction of inner functions in the unit ball of $\mathbb C^d$ is generalized to the space of homogenous type.
The main result of the paper is stated as follows. For every positive continuous function $H$ on the unit sphere $S$ of $\mathbb R^d$ there exists a function $u$ harmonic in the unit ball $B$ of $\mathbb R^d$ such that $\nabla u$ is bounded in $B$ and $|\nabla u|=H$ almost everywhere on $S$.