of the point $x$ under the action of $H$, require that $H(x)$ is uncountable for every $x$ in $X$. If $k := \text{min}\ \{\ \text{dim}\ H(x) :  x\in X\    \}$, then $$\label{comp-emb-2}
L^{p}_{1,H}(X,V_g) \hookrightarrow \hookrightarrow L^q(X,V_g)$$ whenever $1 \leq p < n-k$ and $1 \leq q < \frac{(n-k) p}{n-k-p}$.

In a metric-measure space $(X,d,\mu)$ conditions for the metric $d$ and the measure $\mu$ are sometimes required, leading to *synthetic* extensions of *analytic* Riemannian concepts, like curvature, volume, and dimension (see [@villani] for a friendly introduction to these ideas). We will use the doubling condition for the metric space $(X,d)$ and the lower Ahlfors $s$-regularity of the metric-measure space $(X, d, \mu)$.[^4] For instance, if $(X,g)$ is a compact $n$-manifold with induced distance $d_g$, then $(X,d_g,V_g)$ is a lower $n$-regular metric-measure space, and $(X,d_g)$ is doubling.\
As aforementioned, we use Hajłasz gradients: denote by $M^p_1(X,\mu)$ the vector space of functions in $L^p(X,\mu)$ such that their Hajłasz gradient is also in $L^p(X,\mu)$. In Section \[examples\], Theorem \[riem-mm\], we see that when $(X,d_g, V_g)$ is the natural metric-measure space induced from a compact $n$-manifold $(X,g)$, then $M^p_1(X, V_g)$ coincides with $L^p_1(X, V_g)$ for $1 < p < \infty$.\
In the *analytic* context of 