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.

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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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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/

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