Hausdorff measures and Lebesgue points for the Sobolev classes W ^p _α , α > 0, on spaces of homogeneous type

Abstract

Suppose that (X, µ, d) is a space of homogeneous type, where d is themetric and µ is the measure related by the doubling condition with exponent γ > 0, W ^p _α (X), p > 1, are the generalized Sobolev classes, α > 0, and dim_H is the Hausdorff dimension. We prove that, for any function u ∈ W ^p _α (X), p > 1, 0 < α < γ/p, there exists a set E ⊂ X such that dim_H(E) ≤ γ − αp and, for any x ∈ X ∖ E, the limit

$$ \mathop {\lim }\limits_{r \to + 0} \frac{1} {{\mu (B(x,r))}}\int_{B(x,r)} {ud\mu = u^ * (x)} $$

exists; moreover,

$$ \mathop {\lim }\limits_{r \to + 0} \frac{1} {{\mu (B(x,r))}}\int_{B(x,r)} {|u - u^ * (x)|^q d\mu = 0,} \frac{1} {q} = \frac{1} {p} - \frac{\alpha } {\gamma }. $$

. For α = 1, a similar result was obtained earlier by Hajłasz and Kinnunen in 1998. The case 0 < α ≤ 1 was studied by the author in 2007; in the proof, the structures of the corresponding capacities were significantly used.