Generalized Hajłasz–Sobolev classes on ultrametric measure spaces with doubling condition

We consider the generalized Hajłasz–Sobolev classes $W^p_\alpha (X)$, $\alpha>0$, on ultrametric measure spaces $X$ with doubling condition. We study the massiveness of the complement to the set of Lebesgue points, the convergence rate for Steklov averages, and the problem of Luzin approximation. Bounds for the sizes of exceptional sets are given in terms of capacities.
It is substantial that we remove the constraint $\alpha\le1$ that is necessary for metric spaces. The results of the article were announced in Dokl. Nats. Akad. Nauk Belarusi.