#sobolev space

[mathkonspekt]

This question is connected with my previous: Heisenberg group: function without vertical derivative. Here I am trying to look from another side: what is a difference between Sobolev space and hori...

[mathkonspekt]

Quasicontinuity

from Nonlinear potencial theory..

An important and very useful  property of the Sobolev space $W^m_p(R^N)$ is the fact that every element of this space, viewed as an equivalence class of functions coinciding  a.e., has a distinguished representative which is quasicontinuous with respect to some suitable capacity.

This means, in particular, that the representative is defined more often then almost everywhere with respect to Lebesgue measure, namely, outside a set of capacity 0, where the exceptional sets are measured by Bessel capacity $B_{m, p}$, if $1<p<\infty$, and Hausdorff capacity $\mathcal H^{N-m}_1$, in the case $p=1$ (or by equivalent capacity).

It is thus meaningful to talk about the values of a Sobolev function on "small"  sets with measure 0 but with positive capacity, for instance, on the boundary of some domain; what is meant are just the value of quasicontinuous representative. The property of being quasicontinuous may more generally viewed as the counterpart in potential theory of the familiar Luzin property of measurable functions.