Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical case
Abstract
Let (X, d, µ) be a doubling metric measure space with doubling dimension γ, i. e. for any balls B(x, R) and B(x, r), r < R, following inequality holds µ(B(x, R)) ≤ aµ (R/r)γµ(B(x, r)) for some positive constants γ and aµ. Hajłasz – Sobolev space Mpα(X) can be defined upon such general structure. In the Euclidean case Hajłasz – Sobolev space coincides with classical Sobolev space when p > 1, α = 1. In this article we discuss inclusion of functions from Hajłasz – Sobolev space Mpα(X) into the space of continuous functions for p ≤ 1 in the critical case γ = α p. More precisely, it is shown that any function from Hajłasz – Sobolev class Mpα(X), 0 < p ≤ 1, α > 0, has a continuous representative in case of uniformly perfect space (X, d, µ).
References
- Heinonen J. Nonsmooth calculus. Bulletin of the American mathematical society. 2007;44(2):163–232. DOI: 10.1090/s02730979-07-01140-8.
- Hajłasz P. Sobolev spaces on an arbitrary metric space. Potential Analysis. 1996;5(4):403–415. DOI: 10.1007/BF00275475.
- 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.
- Zhou X. Sobolev functions in the critical case are uniformly continuous in s-Ahlfors regular metric spaces when s ≤ 1. Proceedings of the American Mathematical Society. 2017;145:267–272. DOI: 10.1090/proc/13220.
- Bondarev SA, Krotov VG. Fine properties of functions from Hajłasz – Sobolev classes Mpα, p > 0. I. Lebesgue points. Journal of Contemporary Mathematical Analysis. 2016;51(6):282–295. DOI: 10.3103/S1068362316060029.
- Bondarev SA. Lebesgue points of functions from generalized Sobolev classes Mpα(X) in the critical case. Journal of the Belarusian State University. Mathematics and Informatics. 2018;3:4–11. Russian.
- Górka P, Słabuszewski A. A discontinuous Sobolev function exists. Proceedings of the American Mathematical Society. 2019;147(2):637–639. DOI: 10.1090/proc/14164.
- 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. Burenkov VI, editor. Moscow: Mir; 1973. 342 p.
- Heinonen J. Lectures on analysis on metric spaces. New York: Springer – Verlag; 2001. 141 p. (Universitext). DOI: 10.1007/9781-4613-0131-8.
Copyright (c) 2020 Journal of the Belarusian State University. Mathematics and Informatics
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:
- 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).
- 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.
- 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.)
