Suppose, that there exists a continuous function $f\in C(\mathbb{R}^N)$ with at most linear growth, i.e. $$|f(A)|\leq C(1+|A|)$$ such that the following map (lets call it $F$) $$ \mathbb{B}^N\ni A\mapsto (1-|A|)f\left(\frac{A}{1-|A|}\right) $$ extends to the the boundary $\partial\mathbb{B}^N$, where $\mathbb{B}^N$ is a unit ball in $\mathbb{R}^N$ centered at the origin.
Then the extension $F^{ext}$ can be represented as a limit (fairly easy to check) $$F^{ext}(A):=\lim_{A'\to A, s\to\infty}\frac{f(sA')}{s},$$ where $|A'|<1$ and $A\in\partial\mathbb{B}^N$. Moreover $F^{ext}$ is positively 1-homogeneous, i.e. $F^{ext}(\alpha A)=\alpha F^{ext}(A)$ for $\alpha\geq 0$.
Now, suppose we are given a continuous function $g\in C(\mathbb{R}^N\times\mathbb{R})$ which has the following growth restriction $$|g(D,t)|\leq C(1+|D|+|t|^2)$$ for some $C>0$ and such that the map (lets call it $G$) $$ \mathbb{B}^{N+1}\ni (D,t)\mapsto \left(1+\left|\frac{D}{1-|(D,t)|}\right|+\left|\frac{t}{1-|(D,t)|}\right|^2\right)^{-1}g\left(\frac{D}{1-|(D,t)|},\frac{t}{1-|(D,t)|}\right) $$ extends to the the boundary $\partial\mathbb{B}^{N+1}$. Here $|(D,t)|^2:=|D|^2+t^2$.
I'm curious if there is a way to find a (not necessarily good looking) limit formula for the extension $G^{ext}$ in a similar way to $F^{ext}$, such that $G^{ext}$ is positively (1,2)-homogenous, i.e. $$G^{ext}(\alpha D,\beta\tau)=\alpha\beta^2 G^{ext}(D,\tau)$$ for $\alpha,\beta\geq 0$.
Thank you for any suggestions.