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Register a new userControlled diffeomorphic extension of homeomorphisms
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Let $Omega$ be an internal chord-arc Jordan domain and $varphi:mathbb SrightarrowpartialOmega$ be a homeomorphism. We show that $varphi$ has finite dyadic energy if and only if $varphi$ has a diffeomorphic extension $h: mathbb Drightarrow Omega$ which has finite energy.
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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 integrability of the derivatives of $h$ and double integrals of $varphi$ and of $varphi^{-1} .$
Let $Omegasubseteqmathcal{R}^2$ be a domain, let $X$ be a rearrangement invariant space and let $fin W^{1}X(Omega,mathcal{R}^2)$ be a homeomorphism between $Omega$ and $f(Omega)$. Then there exists a sequence of diffeomorphisms $f_k$ converging to $f$ in the space $W^{1}X(Omega,mathcal{R}^2)$.
We introduce the concept of a new kind of symmetric homeomorphisms on the unit circle, which is derived from the generalization of symmetric homeomorphisms on the real line. By the investigation of the barycentric extension for this class of circle homeomorphisms and the biholomorphic automorphisms induced by trivial Beltrami coefficients, we endow a complex Banach manifold structure on the space of those generalized symmetric homeomorphisms.
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].
The evaluation formula for an elliptic beta integral of type $G_2$ is proved. The integral is expressed by a product of Ruijsenaars elliptic gamma functions, and the formula includes that of Gustafsons $q$-beta integral of type $G_2$ as a special limiting case as $pto 0$. The elliptic beta integral of type $BC_1$ by van Diejen and Spiridonov is effectively used in the proof of the evaluation formula.
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