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Consider a mapping f : X → Y between two metric measure spaces. We study generalized versions of the local Lipschitz number Lip ⁡ f , as well as of the distortion number H f that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for f being a BV mapping f ∈ BV loc ⁢ (X ; Y) or a Newton–Sobolev mapping f ∈ N loc 1 , p ⁢ (X ; Y) , with 1 ≤ p < ∞ . [ABSTRACT FROM AUTHOR]

Given a homeomorphism f:X→Yf:X\to Y between QQ-dimensional spaces X,YX,Y, we show that ff satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that ff belongs to the Sobolev class Nloc1,p(X;Y){N}_{{\rm{loc}}}^{1,p}\left(X;\hspace{0.33em}Y), where 1≤p≤Q1\le p\le Q, and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors QQ-regularity, which in particular enables various weighted spaces to be included in the theory. Notably, even in the classical Euclidean setting, we are able to obtain new results using this approach. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity f∈Nloc1,Q(X;Y)f\in {N}_{{\rm{loc}}}^{1,Q}\left(X;\hspace{0.33em}Y) without the strong assumption of the infinitesimal distortion hf{h}_{f} belonging to L∞(X){L}^{\infty }\left(X).

We study mappings satisfying some estimate of distortion of modulus of families of paths. Under some conditions on definition and mapped domains, we prove that these mappings are logarithmic Hölder continuous at boundary points. [ABSTRACT FROM AUTHOR]

We introduce the critical Besov space B p on quasicircles and prove the solvability of the jump problem on d-regular quasicircles with B p boundary values for d < p < d d - 1 . As applications, we obtain the B p (1 < p < ∞) boundedness of the Cauchy integral operator and the Faber operator on chord-arc curves. [ABSTRACT FROM AUTHOR]

This study defines and describes the properties of the class of diffeomorphisms of the unit disk on the complex plane for which the harmonic measure of the boundary arcs of the slit disk has a limited distortion (i.e., is quasi-invariant). Estimates for derivative mappings of this class are obtained. We prove that such mappings are quasiconformal and quasi-isometries with respect to the pseudohyperbolic metric. An example of a mapping with the specified property is given. As an application, a generalization of the Hayman–Wu theorem to this class of such mappings is proved. [ABSTRACT FROM AUTHOR]

Suppose that f satisfies the following: (1) the polyharmonic equation Δ m f = Δ (Δ m − 1 f) = φ m (φ m ∈ C ( B n ¯ , R n)) , (2) the boundary conditions Δ 0 f = φ 0 , Δ 1 f = φ 1 , ... , Δ m − 1 f = φ m − 1 on S n − 1 ( φ j ∈ C (S n − 1 , R n) for j ∈ { 0 , 1 , ... , m − 1 } and S n − 1 denotes the boundary of the unit ball B n), and (3) f (0) = 0 , where n ≥ 3 and m ≥ 1 are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mappings satisfying the polyharmonic equation with the above Dirichlet boundary value conditions. Furthermore, we establish a Bloch type theorem of mappings satisfying the above polyharmonic equation, which gives an answer to an open problem in Chen and Ponnusamy, (2019). Additionally, we show that if f is a K -quasiconformal self-mapping of B n satisfying the above polyharmonic equation, then f is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as K → 1 + and ‖ φ j ‖ ∞ → 0 + for j ∈ { 1 , ... , m }. [ABSTRACT FROM AUTHOR]

We describe the variation of the Minkowski, packing and Hausdorff dimensions of a set moving under a holomorphic motion, as well as the variation of its area. Our method provides a new, unified approach to various celebrated theorems about quasiconformal mappings, including the work of Astala on the distortion of area and dimension under quasiconformal mappings and the work of Smirnov on the dimension of quasicircles. [ABSTRACT FROM AUTHOR]

On the Riemann sphere, we consider the ptolemaic characteristic of a four of non-empty pairwise non-intersecting compact subsets (generalized tetrad, or generalized angle). We obtain an estimate for distortion of this characteristic under the inverse to a K-quasimeromorphic mapping of the Riemann sphere which takes each of its values at no more then N different points. The distortion function in this estimate depends only on K and N. In the case K=1, it is an essentially new property of complex rational functions.

In an attempt to develop higher-dimensional quasiconformal mappings on metric measure spaces with curvature conditions, i.e. from Ahlfors to Alexandrov, we show that for n≥2 a noncollapsed RCD(0,n) space with Euclidean volume growth is an n-Loewner space and satisfies the infinitesimal-to-global principle.

Suppose that a multi-valued mapping of a domain in the sphere with disjoint images of distinct points boundedly distorts the Ptolemaic characteristic of generalized tetrads (quadruples of disjoint compact sets). Suppose that the image of each has at most components, each of which is a continuum of bounded turning. Then , up to the values at some isolated branch points, is the inverse of a mapping with bounded distortion in the sense of Reshetnyak. In particular, if then the left inverse to is the composition of a quasiconformal automorphism of and a rational function. [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying the distortion of the exterior conformal modulus of a quadriilateral of a sufficiently arbitrary form under the stretching map along the abscissa axis with coefficient . By using the properties of quasiconformal transformations and taking into account some facts from the theory of elliptic integrals, we confirm that the asymptotic behavior of this modulus does not depend on the shape of the boundary of the quadrilateral. Especially, it is equivalent to as . Therefore, we give a solution to the Vuorinen problem for the exterior conformal modulus of a sufficiently arbitrary quadrilateral. [ABSTRACT FROM AUTHOR]

This paper introduces and studies composition operators on the Bergman space with quasiconformal symbols. We first characterize the boundedness of the norms. Then we discuss operator-theoretic issues including compactness characterization, essential norms, Schatten class membership, and conditions for closed ranges. [ABSTRACT FROM AUTHOR]

We study the problem of the existence of a coordinate network of curves on a -smooth bordered surface consisting of geodesic lines covering it up to the set of zero Hausdorff measure and lines orthogonal to them. We additionally assume that the surface is symmetric with respect to some geodesic line. Finding this kind of network is reduced to finding a solution to the nonlinear conjugate Beltrami equation in the unit disk, which degenerates at the points at which the Jacobian of the mapping, carried out by the solution, turns to either zero or infinity. The need for this kind of parametrization is associated with the search for equilibrium forms of liquid droplets in a model that takes into account the presence of an intermediate layer when studying the equilibrium state of a two-phase medium. Using the two-fold method of successive approximations, we establish the presence of a mapping of the unit disk to itself satisfying the Beltrami equation. The constructed mapping is continuous on the closed disk. The real part of the map has the generalized first order derivatives square-integrable on the upper half of the disk and the generalized first order derivatives of the imaginary part are locally square-integrable on it, and the map has the Lusin property. The inverse map, defined almost everywhere on the disk, is almost everywhere injective in it and has the Lusin property. The map found defines an almost global semi-geodesic parametrization of the surface and is a natural generalization of the -quasi-conformal maps introduced by R. Finn and J. Serrin. [ABSTRACT FROM AUTHOR]

In this work, we study the distortion of the exterior conformal modulus of a symmetric quadrilateral, when stretched in the direction of the abscissa axis with the coefficient . By using some facts from the theory of elliptic integrals, we confirm that the asymptotic behavior of this modulus does not depend on the shape of the boundary of the quadrilateral; moreover, it is equivalent to as . [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of derivatives of algebraic polynomials in bounded and unbounded regions of the complex plane. At the same time, both interior and exterior cusp points are allowed on the boundary of such regions. Bernstein-Walsh-type estimates are obtained for derivatives of algebraic polynomials in the specified region with corners for exterior points, as well as Markoff-type estimates for closure of the region. As a result, estimates are found for the derivatives of algebraic polynomials in the whole complex plane. It is also shown that the inequality for the closed region is exact in order for the given region.

In this paper, we study the growth of the mth derivative of an arbitrary algebraic polynomial in bounded and unbounded general domains of the complex plane in weighted Lebesgue spaces. Further, we obtain estimates for the derivatives at the closure of this regions. As a result, estimates for derivatives on the entire complex plane were found.

In this paper, we study the behavior of growth of the modulus of an arbitrary algebraic polynomial in the weighted Bergman space Ap(G, h), p > 0, in exterior points and in closure of region having exterior nonzero and interior zero angles at a finite number of boundary points. Bernstein-Walsh-type estimates are obtained for algebraic polynomials, in bounded regions with a piecewise-smooth boundary. Using an inequality of the Markov-Nikolskii type, the behavior of polynomials in the whole complex plane is given. [ABSTRACT FROM AUTHOR]

In this paper, three equivalent conditions of ρ-harmonic Teichmüller mapping are given firstly. As an application, we investigate the relationship between a ρ-harmonic Teichmüller mapping and its associated holomorphic quadratic differential and obtain a relatively simple method to prove Theorem 2.1 in [1]. Furthermore, the representation theorem of 1/|ω|2-harmonic Teichmüller mappings is given as a by-product. Our results extend the corresponding researches of harmonic Teichmüller mappings.

In this paper, we study Bernstein-Walsh-type estimates for the derivatives of an arbitrary algebraic polynomial on some general regions of the complex plane. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of derivatives of algebraic polynomials in bounded and unbounded regions of the complex plane. At the same time, both interior and exterior cusp points are allowed on the boundary of such regions. Bernstein-Walsh-type estimates are obtained for derivatives of algebraic polynomials in the specified region with corners for exterior points, as well as Markoff-type estimates for closure of the region. As a result, estimates are found for the derivatives of algebraic polynomials in the whole complex plane. It is also shown that the inequality for the closed region is exact in order for the given region. [ABSTRACT FROM AUTHOR]

In this paper, we study the growth of the mth derivative of an arbitrary algebraic polynomial in bounded and unbounded general domains of the complex plane in weighted Lebesgue spaces. Further, we obtain estimates for the derivatives at the closure of this regions. As a result, estimates for derivatives on the entire complex plane were found. [ABSTRACT FROM AUTHOR]

In this note we will consider to what extent some known results on the p -integrable Teichmüller space T p for p ≥ 2 can be extended to the case 1 < p < 2 , giving several new characterizations for p -integrable quasicircles. We will also introduce a new concept T p s which we call the strong p -integrable Teichmüller space and discuss various models of T p s by means of (analytic) Besov space theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Hölder continuity of quasiconformal mappings in ℝ n from the points of view of quasihyperbolic geometry and the theory of Gromov hyperbolic spaces. We establish several characterizations of Gromov hyperbolic domains satisfying the Gehring–Martio-type quasihyperbolic boundary conditions. As applications, we generalize certain results concerning Hölder continuity of conformal mappings, establishing counterparts of results of Becker and Pommerenke, Smith and Stegenga, and Näkki and Palka in higher-dimensional Euclidean spaces. [ABSTRACT FROM AUTHOR]

We describe the asymptotic behavior of the conformal modulus of an unbounded doubly-connected domain, non-symmetric with respect to the coordinate axes, when stretched in the direction of the abscissa axis with coefficient . Therefore, we give a partial answer to a problem, suggested by M. Vourinen, for the case of an arbitrary unbounded domain. [ABSTRACT FROM AUTHOR]

In this paper, we study Bernstein-Walsh type estimates for the higher-order derivatives of an arbitrary algebraic polynomial on quasidisks.

In an attempt to develop higher-dimensional quasiconformal mappings on metric measure spaces with curvature conditions, i.e. from Ahlfors to Alexandrov, we show that for n ě 2 a noncollapsed RCD(0, n) space with Euclidean volume growth is an n-Loewner space and satisfies the infinitesimal-to-global principle. [ABSTRACT FROM AUTHOR]

In the present paper, using the method of boundary integral equations, we consider quasiconformal mappings of closed domains onto a unit circle and establish the estimates allowing one discuss closeness of various values encountered under the construction of these vales through ε, where ε> 0 is a real parameter characterising the domains closeness. In particular, through ε is estimated of the difference of singular integrals corresponding to the close domains and by the order of closeness of domains we estimate the difference of boundary values of mapping functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish several kinds of general Heinz-Schwarz type inequalities for certain self-mappings of unit ball in space. [ABSTRACT FROM AUTHOR]

Let w be a K-quasiconformal self-mapping of the unit disk D satisfying the Dirichlet–Neumann problem: (∂ z ∂ z ¯) 2 w = g in D , w = γ 0 and ∂ ν ∂ z ∂ z ¯ w = γ on T (the unit circle), 1 2 π i ∫ T w ζ ζ ¯ (ζ) d ζ ζ = c , where ∂ ν denotes the differentiation in the outward normal direction. In addition, suppose that the data for w satisfy the following two conditions: 1 2 π ∫ 0 2 π γ (e it) d t = 2 π ∫ D g (ζ) d A (ζ) and w (0) = 0 . The aim of this paper is to prove that w is Lipschitz continuous and, furthermore, it is bi-Lipschitz continuous when |c|, ‖ γ ‖ ∞ and ‖ g ‖ ∞ are small enough. Moreover, the estimates are asymptotically sharp as K → 1 , | c | → 0 , ‖ γ ‖ ∞ → 0 and ‖ g ‖ ∞ → 0 , and so, such a mapping w behaves almost like a rotation for sufficiently small K, |c|, ‖ γ ‖ ∞ and ‖ g ‖ ∞ . [ABSTRACT FROM AUTHOR]

In this paper, some explicit harmonic homeomorphic extensions are constructed. Necessary and sufficient conditions are obtained for the harmonic extensions to be quasiconformal. The extensions considered include those from the unit circle (or real axis) to the unit disk (or upper half plane) and from the unit disk (or upper half plane) to the whole plane. Furthermore, a new concept called close-to-harmonic extension is introduced and several open problems are proposed. [ABSTRACT FROM AUTHOR]

We study the integral characterizations of weighted Hardy spaces of quasiconformal mappings on the n-dimensional unit ball using the weight (1 - r) n - 2 + α . We extend the known results for univalent functions on the unit disk. Some of our results are new even in the unweighted setting for quasiconformal mappings. [ABSTRACT FROM AUTHOR]

An increasing homeomorphism h on the real line R is said to be strongly symmetric if it can be extended to a Quasiconformal homeomorphism of the upper half plane U onto itself whose Bcltrami coefficient µ induces a vanishing Carlcson measure\µ.(z')\2/y dxd.y on U. Wc will deal with the class of strongly symmetric homeomorphisms on the real line and its TcichmuJlcr space, which we call the VMO-Tcichmuller space. In particular, we will show that if h is strongly symmetric on the real line, then it is strongly quasisymmctric such that log h, is a VMO function. This improves some classical results of Carleson (1967) and Anderson-Bcckcr-Lesley (1988) on the problem about the local absolute continuity of a quasisymmetric homeomorphism in terms of the Bcltrami coefficient of a quasiconformal extension. Wc will also discuss various models of the VMO-Teichmiiller space and endow it with a complex Banach manifold structure via the standard Bers embedding. [ABSTRACT FROM AUTHOR]

In the present paper, using the method of boundary integral equations, we consider quasiconformal mappings of close domains onto a unit circle and establish the estimates allowing one to judge on the order of closeness of various values encountered under the construction of these vales through ε, where ε> 0 is a real parameter characterizing the domains closeness. In particular, through ε is defined the order of closeness for the difference of derivatives of solutions of integral equations composed for the close mappable domains by which we construct the desired mapping functions. [ABSTRACT FROM AUTHOR]

In this paper, first we give a new generalization of the Bohr's inequality for the class of bounded analytic functions and for the class of sense-preserving -quasiconformal harmonic mappings of the form where Finally we give a new generalization of the Bohr's inequality for the class of analytic functions subordinate to univalent functions and for the class of sense-preserving -quasiconformal harmonic mappings of the form where is subordinated to some analytic function. [ABSTRACT FROM AUTHOR]

We describe the asymptotic behavior of the conformal modulus of an unbounded doubly connected planar domain, symmetric with respect to the coordinate axes, when stretched in the direction of the abscissa axis with coefficient tending to infinity. Therefore, we give a partial answer to a problem, suggested by M. Vourinen, for the case of unbounded domains. We also use classical theorems by Ahlfors and Warshavskii to investigate the asymptotical behavior of the conformal modulus of a quadrilateral under stretching. [ABSTRACT FROM AUTHOR]

We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to R 2 . Given a measure μ on such a space, we introduce μ -quasiconformal maps f : X → R 2 , whose definition involves deforming lengths of curves by μ . We show that if μ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a μ -quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization. [ABSTRACT FROM AUTHOR]

A relation between 𝜂-quasi-symmetric homomorphisms and K-quasiconformal mappings on n-dimensional smooth connected Riemannian manifolds has been studied. The main results of the research are presented in Theorems 2.6 and 2.7. Several conditions for the boundary behavior of 𝜂-quasi-symmetric homomorphisms between two arbitrary domains with weakly flat boundaries and compact closures, QED and uniform domains on the Riemannian manifolds, which satisfy the obtained results, were also formulated. In addition, quasiballs, c-locally connected domains, and the corresponding results were also considered. [ABSTRACT FROM AUTHOR]

We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

Considering a homeomorphism of the plane onto itself, we prove that the criterion of quasiconformality is the presence of an upper bound for the modulus of the family of curves joining the images of the opposite sides of an arbitrary oriented square with the sides parallel to the coordinate axes. [ABSTRACT FROM AUTHOR]

An explicit example of a Reich sequence for a uniquely extremal quasiconformal mapping in a borderline case between uniqueness and non-uniqueness is given. [ABSTRACT FROM AUTHOR]

We derive a very general condition for a sense-preserving harmonic mapping with dilatation a square to be injective in the unit disk D and to admit a quasiconformal extension to the extended complex plane. The analysis depends on geometric properties of an extension of the Weierstrass–Enneper lift to the extended plane that glues the parametrized minimal surface to a complementary topological hemisphere. The resulting topological sphere renders an entire graph over the complex plane provided additional restrictions on the dilatation are satisfied. The projection results in the desired extension. Several corollaries are drawn from the general criterion. [ABSTRACT FROM AUTHOR]

For two constants K ≥ 1 and K ′ ≥ 0 , suppose that f is a (K , K ′) -quasiconformal self-mapping of the unit disk D , which satisfies the following: (1) the inhomogeneous polyharmonic equation Δ n f = Δ (Δ n - 1 f) = φ n in D (φ n ∈ C (D ¯)) , (2) the boundary conditions Δ n - 1 f = φ n - 1 , ... , Δ 1 f = φ 1 on T ( φ j ∈ C (T) for j ∈ { 1 , ... , n - 1 } and T denotes the unit circle), and (3) f (0) = 0 , where n ≥ 2 is an integer. The main aim of this paper is to prove that f is Lipschitz continuous, and, further, it is bi-Lipschitz continuous when ‖ φ j ‖ ∞ are small enough for j ∈ { 1 , ... , n } . Moreover, the estimates are asymptotically sharp as K → 1 + , K ′ → 0 + and ‖ φ j ‖ ∞ → 0 + for j ∈ { 1 , ... , n } . [ABSTRACT FROM AUTHOR]

The note is devoted to biography and achievements of Ukrainian mathematician Ivan Pesin (1930-1993). [ABSTRACT FROM AUTHOR]