Generalized Poincaré–Sobolev inequality on metric spaces

In this work we prove the inequalities of the form
$$ |f(x_0)-f_{B(x_0,r)}|\le c\eta(r)(\mathcal{S}_\eta f(x_0))^{1-\alpha p/\gamma}\biggl(\,{\int\limits_{B(x_0,r)}\mspace{-31.5mu}{-}\mspace{11.5mu}}(\mathcal{S}_\eta f)^p\,d\mu\biggr)^{\alpha/\gamma} $$
in Lebesgue points of the function $f\in L_{\mathrm{loc}}^1(X)$. Here $0<\alpha<\gamma/p$, $\eta(t)t^{-\alpha}\uparrow$, $\eta(t)t^{-\gamma/p}\downarrow$
$$ \mathcal{S}_\eta f(x)=\sup_{B\ni x}\frac{1}{\eta(r)}{\int\limits_B\hspace{-4.5mm}{-}\mspace{7mu}}|f-f_B|\,d\mu, $$
$B=B(x,r)$ are balls in metric space (or in the homogeneous type space) $X$ with regular Borel measure $\mu$ satisfying the doubling condition of order $\gamma>0$.
We also give some other forms of such inequalities that similar to classic Poincaré inequality and show out their applications to the embedding theorems of the Sobolev type and to the “selfimproving” property of generalizad Poincaré inequality.