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Mat. Zametki, 2007, Volume 82, Issue 1, Pages 99–107 (Mi mz3757)  

This article is cited in 10 scientific papers (total in 10 papers)

Hausdorff Dimension of Lebesgue Sets for $W^p_\alpha$ Classes on Metric Spaces

M. A. Prokhorovich

Belarusian State University

Abstract: Let $(X,\mu,d)$ be a space of homogeneous type, where $d$ and $\mu$ are a metric and a measure, respectively, related to each other by the doubling condition with $\gamma>0$. Let $W^p_\alpha(X)$ be generalized Sobolev classes, let $\operatorname{Cap}_{\alpha,p}$ (where $p>1$ and $0<\alpha\le 1$) be the corresponding capacity, and let $\dim_H$ be the Hausdorff dimension. We show that the capacity $\operatorname{Cap}_{\alpha,p}$ is related to the Hausdorff dimension and also prove that, for each function $u\in W^p_\alpha(X)$, $p>1$, $0<\alpha<\gamma/p$, there exists a set $E\subset X$ such that $\dim_H(E)\le\gamma-\alpha p$, the limit
$$ \lim_{r\to +0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}u d\mu=u^*(x) $$
exists for each $x\in X\setminus E$, and moreover
$$ \lim_{r\to+0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}|u-u^*(x)|^q d\mu=0,\qquad \frac{1}{q}=\frac{1}{p}-\frac{\alpha}{\gamma}. $$


Keywords: Sobolev class, Lebesgue set, capacity, Hausdorff dimension, metric space, Borel measure

DOI: https://doi.org/10.4213/mzm3757

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English version:
Mathematical Notes, 2007, 82:1, 88–95

Bibliographic databases:

UDC: 517.5
Received: 17.05.2006
Revised: 06.12.2006

Citation: M. A. Prokhorovich, “Hausdorff Dimension of Lebesgue Sets for $W^p_\alpha$ Classes on Metric Spaces”, Mat. Zametki, 82:1 (2007), 99–107; Math. Notes, 82:1 (2007), 88–95

Citation in format AMSBIB
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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. G. Krotov, M. A. Prokhorovich, “The Luzin approximation of functions from the classes $W^p_\alpha$ on metric spaces with measure”, Russian Math. (Iz. VUZ), 52:5 (2008), 47–57  mathnet  crossref  mathscinet  zmath  elib
    2. M. A. Prokhorovich, “Hausdorff Measures and Lebesgue Points for the Sobolev Classes $W^p_\alpha$, $\alpha>0$, on Spaces of Homogeneous Type”, Math. Notes, 85:4 (2009), 584–589  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. D. N. Oleshkevich, M. A. Prokhorovich, “Tochki Lebega dlya funktsii iz klassov Soboleva na prostranstve $p$-adicheskikh chisel”, Vestnik BrGU. Seriya 4: Fizika, Matematika, 2010, no. 2, 103–110  hlocal
    4. V. G. Krotov, M. A. Prokhorovich, “The Rate of Convergence of Steklov Means on Metric Measure Spaces and Hausdorff Dimension”, Math. Notes, 89:1 (2011), 156–159  mathnet  crossref  crossref  mathscinet  isi
    5. Veniamin G. Krotov, Mikhail A. Prokhorovich, “Estimates for the Exceptional Lebesgue Sets of Functions from Sobolev Classes”, Recent Advances in Harmonic Analysis and Applications, Springer Proceedings in Mathematics & Statistics, 25, Springer, New York, 2013, 225–234  crossref  mathscinet  zmath  scopus
    6. E. V. Gubkina, K. V. Zabello, M. A. Prokhorovich, E. M. Radyno, “Approksimatsiya Luzina funktsii iz klassov Soboleva na prostranstve mnogomernogo $p$-adicheskogo argumenta”, PFMT, 2013, no. 2(15), 58–65  mathnet
    7. E. V. Gubkina, M. A. Prokhorovich, Ya. M. Radyna, “Generalized Hajłasz–Sobolev classes on ultrametric measure spaces with doubling condition”, Siberian Math. J., 56:5 (2015), 822–826  mathnet  crossref  crossref  isi  elib  elib
    8. Bondarev S.A. Krotov V.G., “Fine properties of functions from Hajłasz–Sobolev classes M p , p 0, I. Lebesgue points”, J. Contemp. Math. Anal.-Armen. Aca., 51:6 (2016), 282–295  crossref  mathscinet  zmath  isi  scopus
    9. Heikkinen T., Koskela P., Tuominen H., “Approximation and quasicontinuity of Besov and Triebel?Lizorkin functions”, Trans. Am. Math. Soc., 369:5 (2017), 3547–3573  crossref  mathscinet  zmath  isi  scopus
    10. Heikkinen T., “Generalized Lebesgue Points For Hajlasz Functions”, J. Funct. space, 2018, 5637042  crossref  mathscinet  zmath  isi  scopus
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