On functions with derivatives in a Lorentz space
The authors of this deep and interesting paper pursue the following question: what are ``minimal'' assumptions on a function \(f\) of several variables which guarantee that \(f\) has properties such as continuity, differentiability, absolute continuity or the Lusin N-property (\(f(E)\) is of \(n\)-dimensional Hausdorff measure zero whenever the same is true for \(E\)). The main result is that one such sharp sufficient condition guaranteeing the N-condition is that \(f\) belongs to the Sobolev space \(W^{1,1}_{\text{loc}}\) and its gradient \(\nabla f\) belongs to the Lorentz space \(L^{n,1}\). Interestingly, the same condition is also sufficient for other important properties such as continuity, \(n\)-absolute continuity or differentiability a.e. In an earlier work of \textit{A. Cianchi} and the reviewer [Ark. Mat. 36, No. 2, 317-340 (1998)] it was shown that the above-mentioned condition is sufficient (and, in a~certain sense, also necessary) for \(f\) to belong to \(L^\infty\). It is therefore interesting and even somewhat surprising that the same condition is equivalent to the \(n\)-absolute continuity of \(f\). The paper contains plenty of nice and original thoughts. A~useful by-product of the argument is a~new characterization of the Lorentz space \(L^{p,q}\) which can turn out to be quite handy.
scientific article; zbMATH DE number 1355508