demands, whenever it is possible, similar studies in this context, see the books [@Ambrosio; @Heinonen; @H].\
A fundamental ingredient in Sobolev spaces is the the concept of weak or distributional gradient. In metric-measure spaces there are at least two notions that provide a valid generalization of the usual gradient in $\mathbb{R}^n$:

-   An upper gradient, see [@Cheeger; @H].

-   The one used in this work, nowadays called a Hajłasz gradient, see [@Ambrosio; @Hajlasz] and Section \[Preliminaries\] below.

Both notions of gradient have advantages and disadvantages with respect to each other, see [@Heinonen; @H; @JSYY].\
If $X$ is a set, $\mu$ is a measure on $X$, and $1 \leq p < \infty$, denote by $L^p(X,\mu)$ the vector space of $\mu$-measurable functions such that $$\| f \|_{L^p(X,\mu)} := (\ \int_X | f(x) |^p d \mu(x)\ )^{1/p}$$ is finite. In particular, if $X$ is either a bounded domain in $\mathbb{R}^n$ or a compact Riemannian $n$-manifold, with $\mu = V_g$ the volume measure associated to the Euclidean or Riemannian line element $g$, respectively, $L^{p}_1(X,\mu)$ refers to the subspace of $L^p(X,\mu)$ made up of those functions such that the norm of their distributional gradient (with respect to $g$) is in $L^p(X,\mu)$. In those cases one 