Mappings of finite distortion: generalized Hausdorff dimension distortion (Q980405)

The authors study how planar maps of finite distortion distort sets of Hausdorff dimension strictly less than two. They measure the image size by means of a generalized Hausdorff measure and prove the following result. Theorem. Let \(f \in W_{\text{loc}}^{1,1}(\Omega; {\mathbb R}^n)\), \( \Omega \subset {\mathbb R}^n\), be a mapping of locally \(\lambda\)-exponentially integrable distortion, \(\lambda > 0\). Set \( h_s(t)= t^2 \log^s(1/t)\) for \(s \in {\mathbb R}\). If \( E \subset {\mathbb R}^2\) satisfies \(\dim_H(E) <2\), then \(H^{h_s}\big(f(E)\big) = 0\) for all \(s < \lambda\), where \(H^{h_s}\) is the generalized Hausdorff measure associated to \(h_s\). They also show that strengthening the integrability hypotheses for \(|Df|\) a stronger conclusion is obtained.