$s$-regular metric-measure space of finite diameter. If $1 < p < s$, then $$\begin{aligned}
        M^p_1(X,\mu)  \hookrightarrow L^{p^*}(X,\mu),
    \end{aligned}$$ where $p^* =\frac{sp}{s-p}$. Moreover, there exists a constant $C=C(s,p,b)$, depending on $s, p, b$, such that for each $f$ in $M^p_1(X,\mu)$ $$\begin{aligned}
        \|f\|_{L^{p^*}(X,\mu)} \leq C\left(\|f\|_{L^{p}(X,\mu)}+\| g_f\|_{L^{p}(X,\mu)}\right)
    \end{aligned}$$ whenever $g_f$ is a Hajłasz gradient for $f$.

We use Proposition \[wlozenie\] to infer:

\[noncritic\] Assume that $(X, d, \mu)$ is an Ahlfors lower $s$-regular compact metric-measure space, with $(X, d)$ doubling. If $1 < p < s$, then for every $q <p^*$ $$\begin{aligned}
        M^p_1(X,\mu)  \hookrightarrow \hookrightarrow L^q(X,\mu),
    \end{aligned}$$ where $p^* =\frac{sp}{s-p}$.

By Proposition \[wlozenie\] have that $M^p_1(X,\mu) \hookrightarrow L^q(X,\mu)$ for every $q \in [1, p^*]$. Moreover, since $(X, d)$ is doubling, by Theorem 2 in [@Kalamajska] we have the compact embedding $$\begin{aligned}
\label{comp}
  M^p_1(X,\mu) \hookrightarrow \hookrightarrow L^{p}(X,\mu).\end{aligned}$$ Hence if $q\in [1,p]$, then $$M^p_1(X,\mu) \hookrightarrow \hookrightarrow L^p(X,\mu)  \hookrightarrow L^q(X,\mu).$$ Next, consider the case when $p < q < p^*$. We will prove that the ball $\mathfrak{B}=\{f: \|f\|_{M^p_1(X,\mu)}\leq 1\}$ is precompact in $L^q(X,\mu)$. Fix $\theta$ in $(0,1)$ so that $$\frac{1}{q}=\frac{\theta}{p}+\frac{1-\theta}{p^*},$$ and use (\[comp\]) to note that $\mathfrak{B}$ is precompact in $L^p(X,\mu)$. Hence for every $\epsilon >0$ there exists an $\tilde{\varepsilon}=2C\epsilon^{\frac{1}{\theta}} / (2C)^{\frac{1}{\theta}}$ net[^8] of $\mathfrak{B}$ in 