Differentiability of monotone Sobolev functions.
The author of this interesting paper studies the question of what are the minimal integrability conditions on the partial derivatives of a~homeomorphic Sobolev mapping from \(W^{1,1}\) that would guarantee differentiability almost everywhere. A~known sufficient condition, due to \textit{E. M. Stein} [Ann.\ Math. (2) 113, 383--385 (1981; Zbl 0531.46021)], is that the gradient \(| \nabla f| \) belongs to the Lorentz space \(L^{n,1}\). A still weaker sufficient condition, due to Väisälä, is that \(| \nabla f| \) is \(p\)-integrable with some \(p>n-1\) (with \(p=n-1\), this does not suffice when \(n>2\)). The author shows that a~natural sufficient condition is \(| \nabla f| \in L^{n-1,1}\), and that this condition implies differentiability almost everywhere for a~fairly large class of mappings. Moreover, he shows that \(| \nabla f| \in L^{n-1,q}\) with any \(q>1\) is no longer sufficient; in such cases, there always exists a~weakly monotone function \(u\) satisfying this condition but not differentiable anywhere.
scientific article; zbMATH DE number 1848876