Geodesic
Inner functions on the spaces of homogeneus type
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 7-14

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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 $S$. 
@article{ZNSL_1983_126_a0,
     author = {A. B. Aleksandrov},
     title = {Inner functions on the spaces of homogeneus type},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {7--14},
     publisher = {mathdoc},
     volume = {126},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_126_a0/}
}
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A. B. Aleksandrov. Inner functions on the spaces of homogeneus type. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XII, Tome 126 (1983), pp. 7-14. http://geodesic.mathdoc.fr/item/ZNSL_1983_126_a0/