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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.
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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 $frac{1}{2}$.
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.
We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudoconvex domains in terms of a density condition or a testing condition on the reproducing kernels. Our methods also yield a sufficient condition for arbitrary domains and lower-dimensional sets.
The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $mathbb R^n$, $nge 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashis pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $mathbb R^n$, this minimal metric coincides with the classical Beltrami-Cayley-Klein metric. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a domain having a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric. Finally, we show that a convex domain is complete hyperbolic if and only if it does not contain any affine 2-plane.
In this paper we introduce a class of pseudo-dissipative holomorphic maps which contains, in particular, the class of infinitesimal generators of semigroups of holomorphic maps on the unit ball of a complex Banach space. We give a growth estimate for maps of this class. In particular, it follows that pseudo-dissipative maps on the unit ball of (infinite-dimensional) Banach spaces are bounded on each domain strictly contained inside the ball. We also present some applications.
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