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Register a new userOn the subinvariance of uniform domains in Banach spaces
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Suppose that $E$ and $E$ denote real Banach spaces with dimension at least 2, that $Dsubset E$ and $Dsubset E$ are domains, and that $f: Dto D$ is a homeomorphism. In this paper, we prove the following subinvariance property for the class of uniform domains: Suppose that $f$ is a freely quasiconformal mapping and that $D$ is uniform. Then the image $f(D_1)$ of every uniform subdomain $D_1$ in $D$ under $f$ is still uniform. This result answers an open problem of Vaisala in the affirmative.
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Suppose that $E$ denotes a real Banach space with the dimension at least 2. The main aim of this paper is to show that a domain $D$ in $E$ is a $psi$-uniform domain if and only if $Dbackslash P$ is a $psi_1$-uniform domain, and $D$ is a uniform domain if and only if $Dbackslash P$ also is a uniform domain, whenever $P$ is a closed countable subset of $D$ satisfying a quasihyperbolic separation condition. This condition requires that the quasihyperbolic distance (w.r.t. $D$) between each pair of distinct points in $P$ has a lower bound greater than or equal to $frac{1}{2}$.
We study the stability of John domains in Banach spaces under removal of a countable set of points. In particular, we prove that the class of John domains is stable in the sense that removing a certain type of closed countable set from the domain yields a new domain which also is a John domain. We apply this result to prove the stability of the inner uniform domains. Finally, we consider a wider class of domains, so called $psi$-John domains and prove a similar result for this class.
We consider a general notion of snowflake of a metric space by composing the distance by a nontrivial concave function. We prove that a snowflake of a metric space $X$ isometrically embeds into some finite-dimensional normed space if and only if $X$ is finite. In the case of power functions we give a uniform bound on the cardinality of $X$ depending only on the power exponent and the dimension of the vector space.
We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincare inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direction of tangent vectors to suitable rectifiable curves.
We prove in the setting of $Q$--Ahlfors regular PI--spaces the following result: if a domain has uniformly large boundary when measured with respect to the $s$--dimensional Hausdorff content, then its visible boundary has large $t$--dimensional Hausdorff content for every $0<t<sleq Q-1$. The visible boundary is the set of points that can be reached by a John curve from a fixed point $z_{0}in Omega$. This generalizes recent results by Koskela-Nandi-Nicolau (from $mathbb{R}^2$) and Azzam ($mathbb{R}^n$). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.
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