Lebesgue points for functions from generalized Sobolev classes Mpa(X) in the critical case

  • Sergey A. Bondarev Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

Abstract

Classical Lebesgue theorem states that for any integrable function almost every point (except the set of measure zero)  is a Lebesgue point. The set of the points that are not Lebesgue points is called an exceptional set. One can estimate the  «size» of the exceptional set for more regular functions (e. g. functions that belong to certain function space) using more  refined than measure characteristics. The paper is devoted to the investigation of the properties of Lebesgue points for  functions from Sobolev classes on general metric space in the critical case γ = α p, γ plays the role of the dimension of the  space, α,  p – smoothness and summability parameters. Estimates of the «size» of the exceptional set in terms of capacities  and Hausdorff dimension are obtained. Exponential rate of convergence for Lebesgue points has been established. Similar  results are known in subcritical case γ > α p as well.

Author Biography

Sergey A. Bondarev, Belarusian State University, Niezaliežnasci Avenue, 4, 220030, Minsk, Belarus

postgraduate student at the department of  function theory, faculty of mechanics and mathematics

References

  1. Stein E. Singular integrals and differentiability properties of functions. Princeton: Princeton University Press; 1970. 287 p. Russian edition: Stein E. Singulyarnye integraly i differentsial’nye svoistva funktsii. Moscow: Mir; 1973. 342 p.
  2. Federer H, Ziemer W. The Lebesgue sets of a function whose distribution derivatives are p­th power summable. Indiana University Mathematics Journal. 1973;22(2):139 –158. DOI: 10.1512/iumj.1973.22.22013.
  3. Bagby T, Ziemer WP. Pointwise differentiability and absolute continuity. Transactions of the American Mathematical Society. 1974;191:129 –148.
  4. Calderón CP, Fabes EB, Riviere NM. Maximal smoothing operators. Indiana University Mathematics Journal. 1974;23: 889 – 898. DOI: 10.1512/iumj.1974.23.23073.
  5. Meyers NG. Taylor expansion of Bessel potentials. Indiana University Mathematics Journal. 1974;23:1043–1049. DOI: 10.1512/iumj.1974.23.23085.
  6. Hajłasz P. Sobolev spaces on an arbitrary metric space. Potential Analysis. 1996;5(4):403– 415. DOI: 10.1007/BF00275475.
  7. Hajłasz P, Kinnunen J. Hölder quasicontinuity of Sobolev functions on metric spaces. Revista Matemática Iberoamericana. 1998;14(3):601– 622. DOI: 10.4171/RMI/246.
  8. Kinnunen J, Latvala V. Lebesgue points for Sobolev functions on metric spaces. Revista Matemática Iberoamericana. 2002; 18(3):685–700. DOI: 10.4171/RMI/332.
  9. Prokhorovich MA. [Capacities and Lebesgue points for fractional Hajłasz – Sobolev classes on metric measure spaces]. Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series. 2006;1:19 –23. Russian.
  10. Prokhorovich MA. [Hausdorff measures and Lebesgue points for Sobolev classes, a > 0, on spaces of homogeneous type]. Matematicheskie zametki [Mathematical Notes]. 2009;85(4):616 – 621. DOI: 10.4213/mzm6642. Russian.
  11. Bondarev SA, Krotov VG. Fine properties of Functions from Hajłasz – Sobolev classes M p a, p > 0. I. Lebesgue points. Journal of Contemporary Mathematical Analysis. 2016;51(6):282–295. DOI: 10.3103/S1068362316060029.
  12. Bondarev SA, Krotov VG. Fine properties of Functions from Hajłasz – Sobolev classes M p a, p > 0. II. Lusin’s approximation. Journal of Contemporary Mathematical Analysis. 2017;52(1):30 –37. DOI: 10.3103/S1068362317010046.
  13. Yang D. New characterization of Hajłasz – Sobolev spaces on metric spaces. Science in China. Series A: Mathematics. 2003; 46(5):675– 689. DOI: 10.1360/02ys0343.
  14. Krotov VG, Porabkovich AI. Estimates of Lp­oscillations of functions for p > 0. Matematicheskie zametki [Mathematical Notes]. 2015;97(3):407– 420. DOI: 10.4213/mzm10600. Russian.
  15. Kinnunen J, Martio O. The Sobolev capacity on metric spaces. Annales Academiæ Scientiarum Fennicæ Mathematica. 1996; 21:367–382.
  16. Heinonen J. Lectures on analysis on metric spaces. Berlin: Springer­Verlag; 2001. 141 p.
  17. Porabkovich AI. [Self­improvement of Lp­Poincare inequality for p > 0]. Chebyshevskii sbornik. 2016;17(1­57):187–200. Russian.
Published
2019-01-19
Keywords: analysis on metric measure spaces, Sobolev spaces, fine properties of functions, Lebesgue points
How to Cite
Bondarev, S. A. (2019). Lebesgue points for functions from generalized Sobolev classes Mpa(X) in the critical case. Journal of the Belarusian State University. Mathematics and Informatics, 3, 4-11. Retrieved from https://journals.bsu.by/index.php/mathematics/article/view/1007
Issue
No 3 (2018): Journal of the Belarusian State University. Mathematics and Informatics
Section
Real, Complex and Functional Analysis

Copyright (c) 2018 Journal of the Belarusian State University. Mathematics and Informatics

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

The authors who are published in this journal agree to the following:

  1. The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
  2. The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
  3. The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)