Geodesic
Inclusion of Hajiasz – Sobolev class $M_p^{\alpha}(X)$ into the space of continuous functions in the critical case
Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2020), pp. 6-12 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $(X, d, \mu)$ be a doubling metric measure space with doubling dimension $\gamma$, i. e. for any balls $B(x, R)$ and $B(x, r), r R$, following inequality holds $\mu(B(x, R)) \leq a_{\mu}(\frac{R}{r})^{\gamma}\mu(B(x, r))$ for some positive constants $\gamma$ and $a_{\mu}$. Hajiasz – Sobolev space $M_p^{\alpha}(X)$ can be defined upon such general structure. In the Euclidean case Hajiasz – Sobolev space coincides with classical Sobolev space when $p > 1,\alpha = 1$. In this article we discuss inclusion of functions from Hajiasz – Sobolev space $M_p^{\alpha}(X)$ into the space of continuous functions for $p \leq 1$ in the critical case $\gamma =\alpha p$. More precisely, it is shown that any function from Hajłasz – Sobolev class $M_p^{\alpha}(X), 0 p \leq 1, \alpha > 0$, has a continuous representative in case of uniformly perfect space $(X, d, \mu)$.
Keywords: analysis on metric measure spaces; Sobolev spaces. 
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     title = {Inclusion of {Hajiasz} {\textendash} {Sobolev} class $M_p^{\alpha}(X)$ into the space of continuous functions in the critical case},
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S. A. Bondarev. Inclusion of Hajiasz – Sobolev class $M_p^{\alpha}(X)$ into the space of continuous functions in the critical case. Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2020), pp. 6-12. http://geodesic.mathdoc.fr/item/BGUMI_2020_1_a0/

[1] J. Heinonen, “Nonsmooth calculus”, Bulletin of the American mathematical society, 44(2) (2007), 163–232 | DOI | MR | Zbl

[2] P. Hajlasz, “Sobolev spaces on an arbitrary metric space”, Potential Analysis, 5(4) (1996), 403–415 | DOI | MR | Zbl

[3] D. Yang, “New characterization of Hajlasz – Sobolev spaces on metric spaces”, Science in China. Mathematics, 46(5) (2003), 675–689 | DOI | MR | Zbl

[4] X. Zhou, “Sobolev functions in the critical case are uniformly continuous in s-Ahlfors regular metric spaces when s = 1”, Proceedings of the American Mathematical Society, 145 (2017), 267–272 | DOI | MR | Zbl

[5] S. A. Bondarev, V. G. Krotov, “Fine properties of functions from Hajlasz – Sobolev classes M (in exp. p, with index alpha), p > 0. I Lebesgue points”, Journal of Contemporary Mathematical Analysis, 51(6) (2016), 282–295 | DOI | MR | Zbl

[6] S. A. Bondarev, “Tochki Lebega dlya funktsii iz obobschennykh klassov Soboleva M (v step. p, s ind. alpha)(X) v kriticheskom sluchae”, Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika, 3 (2018), 4–11 | Zbl

[7] P. Gorka, A. Slabuszewski, “A discontinuous Sobolev function exists”, Proceedings of the American Mathematical Society, 147(2) (2019), 637–639 | DOI | MR | Zbl

[8] I. Stein, “Singulyarnye integraly i differentsialnye svoistva funktsii”, Moskva: Mir, 1973, 342

[9] J. Heinonen, “Lectures on analysis on metric spaces”, New York: Springer – Verlag, 2001, 141 | DOI | MR