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**Notes on absolutely continuous functions of several variables.**
*(English)*
Zbl 1106.26014

The paper contains a variety of interesting observations about absolutely continuous functions of several variables. The point of departure is the theorem of J. Kauhanen, P. Koskela and 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.

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.

Reviewer: Luboš Pick (Praha)

### MSC:

26B30 | Absolutely continuous real functions of several variables, functions of bounded variation |

26B05 | Continuity and differentiation questions |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |