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Matematicheskie Zametki, 2015, Volume 97, Issue 3, Pages 407–420
DOI: https://doi.org/10.4213/mzm10600
(Mi mzm10600)
 

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
Full-text PDF (587 kB) Citations (7)
References:
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
English version:
Mathematical Notes, 2015, Volume 97, Issue 3, Pages 384–395
DOI: https://doi.org/10.1134/S0001434615030098
Bibliographic databases:
Document Type: Article
UDC: 517.5
Language: Russian
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
Citation in format AMSBIB
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\paper Estimates of $L^p$-Oscillations of Functions for $p>0$
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  • This publication is cited in the following 7 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Математические заметки Mathematical Notes
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