Mappings with finite distortion in \(L_{\mathrm{loc}}^p\): modulus of continuity and compression of Hausdorff measure
Let \(\Omega\) be a domain in \(\mathbb{R}^n\). The authors study mappings \(f : \Omega \to \mathbb{R}^n\) such that (i) \(f\) belongs to the Sobolev space \(W_{\mathrm{loc}}^{1,1}(\Omega,\mathbb{R}^n)\), (ii) \(J_f\) belongs to the Lebesgue space \(L_{\mathrm{loc}}^1(\Omega,\mathbb{R}^n)\), (iii) there exists a measurable function \(K : \Omega \to [1,\infty ]\) that is finite almost everywhere and such that, for almost every \(x \in \Omega\), \[ \bigg\| \frac{\partial f_m}{\partial x_k} (x) \bigg\| \leq K(x) J_f (x). \] The function \(K\) is called a distortion function for \(f.\) A non-decreasing function \(h: (0,\infty) \to (0,\infty)\) is called a Hausdorff dimension gauge provided \(\lim _{t\to +0} h(t) =0.\) The authors use a dimension gauge \(h\) to define the Hausdorff measure \(H^h\) via \[ H^h (E) = \lim _{r\to +0}\Big( \inf\Big\{ \sum_i h(\text{diam}\;A_i) : E \subset \cup_i\;A_i;\;\text{diam}\;A_i \leq r \Big\} \Big). \] The authors prove two theorems. { Theorem A}. Let \(\Omega , \Omega^{'} \subset \mathbb{R}^n\) be domains. Suppose \( \Omega @>{f}>> \Omega^{'}\) is a homeomorphism with finite distortion \(K := K_f \in L_{\mathrm{loc}}^p(\Omega)\) for some \(p\in (n-1,\infty)\). Then for each ball \(B:= B(f(z), R)\) with \(\overline{B}\subset \Omega^{'} \) and all points \(x\in G := f^{-1} (B)\subset \Omega,\) \[ |f(x) - f(z)| \geq R \exp \left(- \frac{C(p,n) \| K \|_{L_p(G)}^{1/(n-1)}}{|x-z|^{n/[p(n-1)]}} \right) . \] Moreover, the exponent \(n/[p(n-1)]\) is optimal. { Theorem B}. Let \(p\in (n-1, \infty)\) and \(s\in (0, n].\) Define the Hausdorff gauge function \[ h(t) = h_{s, p, n} (t) := \left( \log\frac{1}{t} \right)^{-ps(n-1)/n}. \] Suppose \(\mathbb{R}^n \supset \Omega @>{f}>> \Omega^{'} \subset \mathbb{R}^n\) is a homeomorphism with finite distortion in \(L_{\mathrm{loc}}^p(\Omega)\). Then for each \(E \subset \Omega \) with \(H^s (E)>0\), \(H^h(f(E)) >0\). They discuss the sharpness of theorem B.
scientific article; zbMATH DE number 6308422