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

V. G. Krotov ; A. I. Porabkovich

Matematičeskie zametki, Tome 97 (2015) no. 3, pp. 407-420

Résumé

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.

Détail

Keywords: $L^p$-oscillations of functions, Sobolev and Hajłasz–Sobolev classes, Poincaré–Sobolev inequalities.
Mots-clés : $\theta$-Lebesgue points