Inclusion of Hajiasz – Sobolev class $M_p^{\alpha}(X)$ into the space of continuous functions in the critical case

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 $(X, d, \mu)$.