and $y$ in $X$. In this context, given $f$ in $M^{p}_1(X,\mu)$, we denote by $g_f$ any Hajłasz gradient for $f$, to endow the space $M^p_1(X,\mu)$ with the norm $$\begin{aligned}
   \label{global-norm}
  \|f\|_{M^p_1(X,\mu)}:= \|f\|_{L^{p}(X,\mu)}+\inf_{g_f} \|g_f\|_{L^{p}(X,\mu)},\end{aligned}$$ and then $M^p_1(X,\mu)$ is a Banach space.

In the same context, say that $f \in L^p(X,\mu)$ belongs to $m^p_1(X,\mu)$ if there exists some $g \in L^{p}(X,\mu)$, called a *local Hajłasz gradient*, such that for every $z$ in $X$ there exists an open set $U_z$ and some $E_z \subset U_z$ with $\mu(E_z) = 0$, such that for every pair of points $\{x,y\}$ in $U_z  \thicksim E_z$ the inequality (\[global-local\]) holds. As in (\[global-norm\]) one defines $$\begin{aligned}
   \label{local-norm}
  \|f\|_{m^p_1(X,\mu)}:= \|f\|_{L^{p}(X,\mu)}+\inf_{g_f} \|g_f\|_{L^{p}(X,\mu)},\end{aligned}$$ where now the infimum is over all those $g_f$ that are local Hajłasz gradients for $f$. Then $m^p_1(X,\mu)$ is also a Banach space. It is obvious that Hajłasz gradients for a function $f$ are local Hajłasz gradients; the converse is not true in general, see [@JSYY] for an example.

For a detailed exposition of some basic properties of these spaces, we refer to [@Ambrosio; @Hajlasz; @Hajlasz1; @HajlaszKoskela; @Heinonen; @H; @JSYY; @Kinnunen].\
The next result will be useful:

(See [@Hajlasz]) \[wlozenie\] Suppose $(X, d, \mu)$ is an Ahlfors lower 