Suppose that $(X,\mu,d)$ and $(Y,\nu,\rho)$ are doubling metric measure spaces.  Fix $\alpha>0$ and define the space, analogously [to this paper][1], as the collection of all measurable functions $f:X\rightarrow Y$ satisfying:
$$
\left(
\int_0^{\infty}\left[\int_{y \in Q}\int_{x \in Q} \frac{\rho(f(x),f(y))^p}{\mu(B(x,t))^{\alpha}} d\mu(x)d\mu(y)\right]^{\frac{p}{q}} \frac1{t^{1+sq}} dt
\right)^q
< \infty
$$

Then the functions satisfying the above constraint can be seen as a non-Euclidean analogue of Hajłasz-Besov spaces, similar to the Korevaar-Shoen extensions of the Sobolev space as seen [in this paper][2].  My question is:

 - Are these objects studied?  If so what are some key papers?  

 - More interestingly, is the subset of *continuous* functions satisfying the above relation studied?


  [1]: https://www.degruyter.com/downloadpdf/j/form.2013.25.issue-4/form.2011.135/form.2011.135.pdf
  [2]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.495.1895&rep=rep1&type=pdf