Frequency of Sobolev and quasiconformal dimension distortion
The authors study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. \textit{P.~Hajłasz} and \textit{J.~Tyson} constructed [Mich. Math. J. 56, No. 3, 687--702 (2008; Zbl 1165.28004)] Sobolev mappings in the Reshetnyak-Sobolev space \({\mathbb R}^{1,n}\) from the \(n\)-dimensional cube onto any length-compact space, a class which includes certain infinite-dimensional spaces. On the other hand, it follows from the Sobolev embedding theorem that, in the case a Sobolev exponent \(p\) larger than \(n\), the dimension of the image is at most \(n\). (When \(p>n\), we always talk of the continuous representative.) However, subsets of the cube that have dimension less than \(n\) may still suffer an increase in dimension. \textit{R.~Kaufman} obtained [Proc. Am. Math. Soc. 128, No. 2, 427--431 (2000; Zbl 0938.28001)] better bounds than one would deduce by simply applying the Hölder continuity arising from the embedding theorem. The following version of one of Kaufman's results is the starting point for the paper under review. Denoting by \(G(n,m)\) the Grassmannian, by \(V^\perp\) the orthogonal complement of \(V\in G(n,m)\), and for \(a\in V^\perp\) setting \(V_a=V+a\): Let \(f\in W^{1,p}(\Omega,Y)\) for \(p>n\) and let \(V\) in the Grassmannian \(G(n,m)\). Then \(f(V_a\cap \Omega)\) has zero \({\mathcal{H}}^{pm/(p-n+m)}\) measure for each \(a\in V^\perp\), in particular, \[ \dim f(V_a\cap \Omega)\leq {pm\over p-n+m}. \] The authors generalize this result as follows by allowing an exceptional set in \(V^\perp\). For integers \(m\) and \(n\) satisfying \(1\leq m\leq n-1\), for \(p\geq 1\) and \(m\leq \alpha\leq {pm\over p-n+m}\), they set \[ \beta(p,\alpha):=(n-m)-(1-{m\over\alpha})p. \] Suppose \(\Omega\) in \(\mathbb R^n\) is a domain, \(f\in W^{1,p}(\Omega,Y)\), \(p>n\), \(V\in G(n,m)\), and \[ m<\alpha\leq {pm\over p-n+m}. \] Then \(f(V_\alpha\cap \Omega)\) has zero \({\mathcal H}^\alpha\) measure for \({\mathcal H}^\beta\)-almost every \(a\in V^\perp\), where \(\beta=\beta(p,\alpha)\). The authors show the sharpness of their result by constructing a \(W^{1,p}\)-map that increases the dimension of each element in a \(\beta(p,\alpha)\)-dimensional set of parallel affine \(m\)-dimensional subspaces of \(\mathbb R^n\) from \(m\) to \(\alpha\). As an application of the positive result, quasiconformal mappings are considered. They are a special class of mappings falling into the scope of the theorem. The authors also provide examples of quasiconformal maps that map many subspaces onto sets of high Minkowski dimension. It is interesting to note that the examples showing the sharpness in the Sobolev case are not explicitly constructed. The statement is deduced by showing that the expectation of some desired quantity is finite. The authors remark that they do not know whether a deterministic construction can be given. The quasiconformal examples are however concretely constructed. The paper concludes with a section in which open problems and questions are presented. Readers interested in the paper under review might also want to have a look at the paper by \textit{S.~Hencl} and \textit{P.~Honzík} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29, No. 3, 401--411 (2012; Zbl 1245.28006)]. In their article, the case \(p<n\) is treated.
scientific article; zbMATH DE number 6137364