We assume that the measure of every open non-empty set is positive, and that the measure of every bounded set is finite.

In most parts of our paper we will assume that the metric-measure space $(X, d, \mu)$ is *lower Ahlfors $s$-regular*: this means that there exists a constant $b$ such that $$b R^s \leq \mu \left(B_R(x)\right)$$ for all balls $B_R(x)$ in $X$ with $R < \hbox{diam} X$.

A metric space is said to be *doubling* if there exists a constant $C$ such that for every ball of radius $R$, there exist $C$ balls of radius $R/2$ that cover the original ball. It not difficult to see that if $(X, d, \mu)$ is a doubling metric-measure space,[^7] then $(X, d)$ is a doubling metric space (see [@gromov], Appendix $B_+$). Conversely, J. Luukkainen and E. Saksman in [@Luukkainen] prove that every complete doubling metric space carries a doubling measure.\
If $(X, d, \mu)$ is a metric-measure space, say that a function $f$ in $L^p(X,\mu)$ belongs to the *Haj[ł]{}asz-Sobolev* space $M^{p}_{1}(X,\mu)$ if there exists some $g \in L^{p}(X,\mu)$, called a *Hajłasz gradient*, such that $$\begin{aligned}
   \label{global-local}
|f(x) - f(y)| \leq  d(x,y) \left( g(x) + g(y) \right) \end{aligned}$$ for $\mu$ almost every $x$ 