Notes on absolutely continuous functions of several variables (Q1772920)

The paper contains a~variety of interesting observations about absolutely continuous functions of several variables. The point of departure is the theorem of \textit{J. Kauhanen, P. Koskela} and \textit{J. Malý} [Manuscr. Math. 100, No. 1, 87--101 (1999; Zbl 0976.26004)], which states that any function on a~domain \(\Omega\subset\mathbb R^ n\) whose gradient belongs to the Lorentz space \(L^{n,1}\) has an~absolutely continuous representative. Let us recall that this is a~fairly sharp result in view of the known fact that \(L^{n,1}\) is the optimal such a thing, that is, the largest possible rearrangement-invariant Banach function space such that the first-order Sobolev space built upon it is continuously embedded into \(C\) or \(L^{\infty}\). The first main result of the paper shows, via an~ingenious analysis of rearrangements, that there is an~absolutely continuous function on a~ball in \(\mathbb R^ n\), whose gradient lies outside the Lorentz space \(L^{2,1}\). Interestingly, this function is radially decreasing and its `radial envelope' near the origin is governed by \(\frac {r\sin(1/r)}{\log r}\). Furthermore, the author gives a~number of results concerning fine properties of absolutely continuous functions near the boundary of \(\Omega\). The information is split into a~section of `bad news' and another one of `good ones'. The negative side of things goes first, as usual; here the author points out, for instance, that there exist domains \(\Omega\) bad enough to allow absolutely continuous functions not to have a~continuous extension to the boundary. In fact, even domains with boundary as good as \(C^{1,\alpha}\) with \(\alpha\) strictly less than one are not good enough. On the other hand, if the quality of the boundary of \(\Omega\) is \(C^{1,1}\) (or better), then the continuous extension exists.