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?


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