# Fractional Hajłasz-Besov-like similar to the Korevaar-Schoen-Sobolev spaces?

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, 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. 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?