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Let $mathbb{X}$ be a Jordan domain satisfying hyperbolic growth conditions. Assume that $varphi$ is a homeomorphism from the boundary $partial mathbb{X}$ of $mathbb{X}$ onto the unit circle. Denote by $h$ the harmonic diffeomorphic extension of $varphi $ from $mathbb{X}$ onto the unit disk. We establish the optimal Orlicz-Sobolev regularity and weighted Sobolev estimate of $h.$ These generalize the Sobolev regularity of $h$ by Koski-Onninen [21, Theorem 3.1].
Let $Omega subset mbr^2$ be an internal chord-arc domain and $varphi : mbs^1 rightarrow partial Omega$ be a homeomorphism. Then there is a diffeomorphic extension $h : mbd rightarrow Omega$ of $varphi .$ We study the relationship between weighted int
We obtain explicit and simple conditions which in many cases allow one decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As
We obtain sharp ranges of $L^p$-boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$-bounde
It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $mathcal C^1$ boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. Th
The conformal mapping $f(z)=(z+1)^2 $ from $mathbb{D}$ onto the standard cardioid has a homeomorphic extension of finite distortion to entire $mathbb{R}^2 .$ We study the optimal regularity of such extensions, in terms of the integrability degree of