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This article is cited in 7 scientific papers (total in 7 papers)
Estimates of $L^p$-Oscillations of Functions for $p>0$
V. G. Krotov, A. I. Porabkovich Belarusian State University
Abstract:
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.
Keywords:
$L^p$-oscillations of functions, $\theta$-Lebesgue points, Sobolev and Hajłasz–Sobolev classes,
Poincaré–Sobolev inequalities.
Received: 19.06.2014 Revised: 22.10.2014
Citation:
V. G. Krotov, A. I. Porabkovich, “Estimates of $L^p$-Oscillations of Functions for $p>0$”, Mat. Zametki, 97:3 (2015), 407–420; Math. Notes, 97:3 (2015), 384–395
Linking options:
https://www.mathnet.ru/eng/mzm10600https://doi.org/10.4213/mzm10600 https://www.mathnet.ru/eng/mzm/v97/i3/p407
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Abstract page: | 587 | Full-text PDF : | 153 | References: | 120 | First page: | 75 |
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