Fine properties of functions from Hajłasz–Sobolev classes Mαp, p > 0, II. Lusin’s approximation

Abstract

The present paper is devoted to the Lusin’s approximation of functions from Hajłasz–Sobolev classes M_α^p(X) for p > 0. It is proved that for any f ∈ M_α^p(X) and any ε > 0 there exist an open set O_ε ⊂ X with measure less than ε (as a measure can be taken the corresponding capacity or Hausdorff content) and an approximating function f_ε such that f = f_ε on XO_ε. Moreover, the correcting function f_ε is regular (that is, it belongs to the underlying space M_α^p(X) and it is a locally Hölder function), and it approximates the original function in the metric of the space M_α^p(X).