Estimates of $L^p$-Oscillations of Functions for $p>0$

We prove a number of inequalities for the mean oscillations
$$ \mathcal{O}_{\theta}(f,B,I)=\biggl(\frac{1}{\mu(B)} \int_B |f(y)-I|^\theta\,d\mu(y)\biggr)^{1/\theta}, $$
where $\theta>0$, $B$ is a ball in a metric space with measure $\mu$ satisfying the doubling condition, and the number $I$ is chosen in one of the following ways: $I=f(x)$ ($x\in B$), $I$ is the mean value of the function $f$ over the ball $B$, and $I$ is the best approximation of $f$ by constants in the metric of $L^{\theta}(B)$. These inequalities are used to obtain $L^p$-estimates ($p>0$) of the maximal operators measuring local smoothness, to describe Sobolev-type spaces, and to study the self-improvement property of Poincaré–Sobolev-type inequalities.