Hausdorff Measures and Lebesgue Points for the Sobolev Classes~$W^p_\alpha$, $\alpha>0$, on Spaces of Homogeneous Type

M. A. Prokhorovich

Matematičeskie zametki, Tome 85 (2009) no. 4, pp. 616-621

Résumé

Suppose that $(X,\mu,d)$ is a space of homogeneous type, where $d$ is the metric and $\mu$ is the measure related by the doubling condition with exponent $\gamma>0$, $W^p_\alpha(X)$, $p>1$, are the generalized Sobolev classes, $\alpha>0$, and $\operatorname{dim_H}$ is the Hausdorff dimension. We prove that, for any function $u\in W^p_\alpha(X)$, $p>1$, $0\alpha\gamma/p$, there exists a set $E\subset X$ such that $\operatorname{dim_H}(E)\le\gamma-\alpha p$ and, for any $x\in X\setminus E$, the limit $$ \lim_{r\to+0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}u\,d\mu=u^{*}(x) $$ exists; moreover, $$ \lim_{r\to+0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}|u-u^{*}(x)|^{q}\,d\mu=0,\qquad \frac{1}{q}=\frac{1}{p}-\frac{\alpha}{\gamma}. $$ For $\alpha=1$, a similar result was obtained earlier by Hajłasz and Kinnunen in 1998. The case $0\alpha\le1$ was studied by the author in 2007; in the proof, the structures of the corresponding capacities were significantly used.

Détail

Keywords: Hausdorff measure, Sobolev classes $W^p_\alpha$, Hölder classes $H^\alpha(X)$, Borel measure, Hausdorff capacity
Mots-clés : Lebesgue point, $\alpha>0$, Hausdorff dimension.