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Register a new userTeichmullers problem in space
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Quasiconformal homeomorphisms of the whole space Rn, onto itself normalized at one or two points are studied. In particular, the stability theory, the case when the maximal dilatation tends to 1, is in the focus. Our main result provides a spatial analogue of a classical result due to Teichmuller. Unlike Teichmullers result, our bounds are explicit. Explicit bounds are based on two sharp well-known distortion results: the quasiconformal Schwarz lemma and the bound for linear dilatation. Moreover, Bernoulli type inequalities and asymptotically sharp bounds for special functions involving complete elliptic integrals are applied to simplify the computations. Finally, we discuss the behavior of the quasihyperbolic metric under quasiconformal maps and prove a sharp result for quasiconformal maps of R^n {0} onto itself.
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For a given ring (domain) in $overline{mathbb{R}}^n$ we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all $nge 3,,$ the standard definition of uniformly perfect sets in terms of Euclidean metric is equivalent to the boundedness of moduli of separating rings. We also establish separation theorems for a half of a ring. As applications of those results, we will prove boundary Holder continuity of quasiconformal mappings of the ball or the half space in $mathbb{R}^n.$
The Darlington synthesis problem (in the scalar case) is the problem of embedding a given contractive analytic function to an inner $2times 2$ matrix function as the entry. A fundamental result of Arov--Douglas--Helton relates this algebraic property to a pure analytic one known as a pseudocontinuation of bounded type. We suggest a local version of the Darlington synthesis problem and prove a local analog of the ADH theorem.
It was known to von Neumann in the 1950s that the integer lattice $mathbb{Z}^2$ forms a uniqueness set for the Bargmann-Fock space. It was later demonstrated by Seip and Wallsten that a sequence of points $Gamma$ that is uniformly close to the integer lattice is still a uniqueness set. We show in this paper that the uniqueness sets for the Fock space are preserved under much more general perturbations.
A description of the Bloch functions that can be approximated in the Bloch norm by functions in the Hardy space $H^p$ of the unit ball of $Cn$ for $0<p<infty$ is given. When $0<pleq1$, the result is new even in the case of the unit disk.
We show that for an entire function $varphi$ belonging to the Fock space ${mathscr F}^2(mathbb{C}^n)$ on the complex Euclidean space $mathbb{C}^n$, the integral operator begin{eqnarray*} S_{varphi}F(z)=int_{mathbb{C}^n} F(w) e^{z cdotbar{w}} varphi(z- bar{w}),dlambda(w), zin mathbb{C}^n, end{eqnarray*} is bounded on ${mathscr F}^2(mathbb{C}^n)$ if and only if there exists a function $min L^{infty}(mathbb{R}^n)$ such that $$ varphi(z)=int_{mathbb{R}^n} m(x)e^{-2left(x-frac{i}{2} z right)cdot left(x-frac{i}{2} z right)} dx, zin mathbb{C}^n. $$ Here $dlambda(w)= pi^{-n}e^{-leftvert wrightvert^2}dw$ is the Gaussian measure on $mathbb C^n$. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_varphi$. Moreover, we obtain the reducing subspaces of $S_{varphi}$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${mathscr F}^2(mathbb{C})$ on the complex plane ${mathbb C}$ (Integr. Equ. Oper. Theory {bf 81} (2015), 451--454).
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