Hausdorff Dimension of Lebesgue Sets for $W^p_\alpha$ Classes on Metric Spaces

M. A. Prokhorovich

Matematičeskie zametki, Tome 82 (2007) no. 1, pp. 99-107

Résumé

Let $(X,\mu,d)$ be a space of homogeneous type, where $d$ and $\mu$ are a metric and a measure, respectively, related to each other by the doubling condition with $\gamma>0$. Let $W^p_\alpha(X)$ be generalized Sobolev classes, let $\operatorname{Cap}_{\alpha,p}$ (where $p>1$ and $0\alpha\le 1$) be the corresponding capacity, and let $\dim_H$ be the Hausdorff dimension. We show that the capacity $\operatorname{Cap}_{\alpha,p}$ is related to the Hausdorff dimension and also prove that, for each function $u\in W^p_\alpha(X)$, $p>1$, $0\alpha\gamma/p$, there exists a set $E\subset X$ such that $\dim_H(E)\le\gamma-\alpha p$, the limit $$ \lim_{r\to +0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}u\,d\mu=u^*(x) $$ exists for each $x\in X\setminus E$, and moreover $$ \lim_{r\to+0}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}|u-u^*(x)|^q\,d\mu=0,\qquad \frac{1}{q}=\frac{1}{p}-\frac{\alpha}{\gamma}. $$

Détail

Keywords: Sobolev class, capacity, metric space, Borel measure.
Mots-clés : Lebesgue set, Hausdorff dimension