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Wall properties of domains in euclidean spaces. (English) Zbl 1017.30029

For \(n\geq 2\), \(1\leq p\leq n-1\) and \(c\geq 1\), let \(G\subset \mathbb{R}^n\) be a domain with the properties:
(1) every map \(f: S^{n-1}\to G\) is null-homotopic;
(2) if \(x\in \mathbb{R}^n\), \(t>0\), \(n-p\leq k\leq n-2\), then every map \(f: S^k\to G\cap B(x,t)\) is null-homotopic in \(G\cap B(x,ct)\) where \(B(x,r)\) is an open ball in \(\mathbb{R}^n\) centered at \(x\) and of radius \(r\). Let \(m_p\) and \(m_p^{\infty}\) be the Hausdorff \(p\)-measure and the Hausdorff \(p\)-content respectively. The author proved earlier that for arbitrary \(a\in G\) the inequality \[ m_p^{\infty}(\partial G\cap B(a,2r)\geq r^p/c_1 \] holds with \(c_1=c_1(c,n,p)\) and \(r=d(a,\partial G)\). This is a homotopical version implying the Heinonen quasiconformal wall conjecture: \[ m_{n-1}(\partial G\cap B(a,2r))\geq r^{n-1}/c \] where a domain \(G\) is \(K\)-quasiconformally equivalent to a ball and \(c=c(K,n)\). The author also has proved the homological version requiring that the above properties (1) and (2) are replaced by
(1’) \(H_{n-1}(G)=0\);
(2’) the natural homomorphisms \(H_k(G\cap B(x,t))\to H_k(G\cap B(x,ct))\) are zero for \(n-p\leq k\leq n-2\).
In the present paper the author proves that the homological, homotopical and the quasiconformal wall theorems are true with \(m_p^{\infty}\) replaced by the projectional \(p\)-measure \(\mu_p\) where, for \(A\subset \mathbb{R}^n\) and the orthogonal projection \(\pi_E: \mathbb{R}^n\to E\), \( \mu_p(A)=\sup\{m_p(\pi_EA): E \) is a \(p\) -dimensional linear subspace of \(\mathbb{R}^n\}\). The \((c,p)\)-wall property, \(c>0\), \(1\leq p\leq n-1\), of a domain \(G\subset \mathbb{R}^n\) is defined by the inequality \[ \mu_p(\partial G\cap B(a,2r))\geq r^p/c, a\in G. \] The inner \((c,p)\)-wall property of \(G\) is defined by \[ \mu_p(\partial G\cap B_{\lambda}(a,2r))\geq r^p/c, a\in G, \] where \(B_{\lambda}(a,r)\) denotes the set of all \(x\in\overline G\) such that \(\lambda(a,x)=\inf_{\gamma}l(\gamma)<r\) and the infimum of the length \(l(\gamma)\) of \(\gamma\) is taken over all rectifiable arcs \(\gamma\) joining \(a\) and \(x\) in \(G\). The author proves that if a domain \(G\) has the \((c,p)\)-wall property, then it has the inner \((c',p)\)-wall property with \(c'=c'(c,p)\). This implies that if \(G\) satisfies the hypotheses of one of the author’s wall theorems with a constant \(c>0\) and an integer \(p\in[1,n-1]\), then \(G\) has the inner \((c',p)\)-wall property with \(c'=c'(c,p,n)\). If \(G\) is \(K\)-quasiconformally equivalent to a ball, then \(G\) has the inner \((c',n-1)\)-wall property with \(c'=c'(K,n)\).

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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