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Mat. Zametki, 2015, Volume 97, Issue 3, Pages 407–420 (Mi mz10600)  

This article is cited in 4 scientific papers (total in 4 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
$$ \mathcal{O}_{\theta}(f,B,I)=(\frac{1}{\mu(B)} \int_B |f(y)-I|^\theta d\mu(y))^{1/\theta}, $$
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.

DOI: https://doi.org/10.4213/mzm10600

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English version:
Mathematical Notes, 2015, 97:3, 384–395

Bibliographic databases:

Document Type: Article
UDC: 517.5
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

Citation in format AMSBIB
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\by V.~G.~Krotov, A.~I.~Porabkovich
\paper Estimates of $L^p$-Oscillations of Functions for $p>0$
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\issue 3
\pages 407--420
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. D. Stepanov, “On Optimal Banach Spaces Containing a Weight Cone of Monotone or Quasiconcave Functions”, Math. Notes, 98:6 (2015), 957–970  mathnet  crossref  crossref  mathscinet  isi  elib
    2. A. I. Porabkovich, “Samouluchshenie $L^p$-neravenstva Puankare pri $p>0$”, Chebyshevskii sb., 17:1 (2016), 187–200  mathnet  elib
    3. S. A. Bondarev, V. G. Krotov, “Fine properties of functions from Hajłasz–Sobolev classes $M_\alpha^p$, $p>0$, I. Lebesgue points”, J. Contemp. Math. Anal., 51:6 (2016), 282–295  crossref  mathscinet  zmath  isi  scopus
    4. S. A. Bondarev, V. G. Krotov, “Fine properties of functions from Hajłasz–Sobolev classes $M_\alpha^p$, $p>0$, II. Luzin approximation”, J. Contemp. Math. Anal., 52:1 (2017), 30–37  crossref  mathscinet  zmath  isi  scopus
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