Fine properties of functions from Hajłasz–Sobolev classes Mαp, p > 0, I. Lebesgue points

Abstract

Let X be a metric measure space satisfying the doubling condition of order γ > 0. For a function f ∈ L_loc^p(X), p > 0 and a ball B ⊂ X by I_B^(p)f we denote the best approximation by constants in the space L^p(B). In this paper, for functions f from Hajłasz–Sobolev classes M_α^p(X), p > 0, α > 0, we investigate the size of the set E of points for which the limit lim_r→+0I_B(x,r)^(p)f = f*(x). exists. We prove that the complement of the set E has zero outer measure for some general class of outer measures (in particular, it has zero capacity). A sharp estimate of the Hausdorff dimension of this complement is given. Besides, it is shown that for x ∈ E

\(\mathop {\lim }\limits_{r \to + 0} {\int_{B\left( {x,r} \right)} {\left| {f - f*\left( x \right)} \right|} ^q}d\mu = 0,{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q} = {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p} - {\alpha \mathord{\left/ {\vphantom {\alpha r}} \right. \kern-\nulldelimiterspace} r}.\)

Similar results are also proved for the sets where the "means" I_B(x,r)^(p)f converge with a specified rate.