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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