An example concerning the zero set of the Jacobian

Let f∈W1,1(Ω,Rn) be a homeomorphism of finite distortion K. It is known that if K1/(n−1)∈L¹(Ω), then the Jacobian Jf of f is positive almost everywhere in Ω. We will show that this integrability assumption on K is sharp in any Orlicz-scale: if α is increasing function (satisfying minor technical assumptions) such that limt→∞α(t)=∞, then there exists f such that K1/(n−1)/α(K)∈L¹(Ω) and Jf vanishes in a set of positive measure.     

Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.     

Quasiconformal homogeneity of genus zero surfaces

A Riemann surface M is said to be K-quasiconformally homogeneous if, for every two points p, q ∈ M, there exists a K-quasiconformal homeomorphism f: M→M such that f(p) = q. In this paper, we show there exists a universal constant K > 1 such that if M is a K-quasiconformally homogeneous hyperbolic genus zero surface other than ⅅ², then K ≥ K. This answers a question by Gehring and Palka [10]. Further, we show that a non-maximal hyperbolic surface of genus g ≥ 1 is not K-quasiconformally homogeneous for any finite K ≥ 1.     

Local stability of mappings with bounded distortion on Heisenberg groups

This is the second of the author’s three papers on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K ? 1 in the Sobolev norm L _p ¹ for all \(p < \tfrac{C}{{K - 1}}\).In this paper we prove a local variant of the desired result: each mapping on a ball with bounded distortion and distortion coefficient K near to 1 is close on a smaller ball to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K ? 1 in the Sobolev norm L _p ¹ for all \(p < \tfrac{C}{{K - 1}}\). We construct an example that demonstrates the asymptotic sharpness of the order of closeness of a mapping with bounded distortion to a conformal mapping in the Sobolev norm.     

Mappings of Finite Distortion of Polynomial Type

Suppose f:? ⁿ →? ⁿ is a mapping of K-bounded p-mean distortion for some p>n?1. We prove the equivalence of the following properties of f: the doubling condition for J(x,f) over big balls centered at the origin, the boundedness of the multiplicity function N(f,? ⁿ ), the polynomial type of f, and the polynomial growth condition for f.     

Stability of mappings with bounded distortion in the Sobolev norm on the John domains of Heisenberg groups

This article completes the authors’s series on stability in the Liouville theorem on the Heisenberg group. We show that every mapping with bounded distortion on a John domain of the Heisenberg group is approximated by a conformal mapping with order of closeness √K ? 1 in the uniform norm and with order of closeness K ? 1 in the Sobolev L _p ¹ -norm for all p < C/K?1. We construct two examples, demonstrating the asymptotic sharpness of our results.     

Quasiconformality of the injective mappings transforming spheres to quasispheres

We prove that every injective mapping of a domain \(D \subset \overline {{\mathbb{R}^n}} \) transforming spheres Σ ? D to K-quasispheres (the images of spheres under K-quasiconformal automorphisms of \(\overline {{\mathbb{R}^n}} \)) is K′-quasiconformal with K′ depending only on K and tending to 1 as K → 1. This is a quasiconformal analog of the classical Carathéodory Theorem on the Möbius property of an injective mapping of a domain D ? Rⁿ which sends spheres to spheres.     

The class of mappings with bounded specific oscillation,and integrability of mappings with bounded distortion on Carnot groups

This paper is the first of the author’s three articles on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K ? 1 in the Sobolev norm L _p ¹ for all \(p < \tfrac{C}{{K - 1}}\).In the present article we study integrability of mappings with bounded specific oscillation on spaces of homogeneous type. As an example, we consider mappings with bounded distortion on the Heisenberg group. We prove that a mapping with bounded distortion belongs to the Sobolev class W _p,loc ¹ , where p → ∞ as the distortion coefficient tends to 1.     

Radial Limits of Mappings of Bounded and Finite Distortion

We give sufficient conditions for mappings defined on the unit ball of ? ⁿ to have radial limits almost everywhere. In particular, we show that if f:B(0,1)→? ⁿ is a mapping with exponentially integrable distortion satisfying the growth condition $$\int_{B(0,r)}J_f(x)\,dx\leq c(1-r)^{-a} $$ for some a∈[0,n?1), then . Here the set E(f) consists of those points in ?B(0,1) where f does not have radial limits. We also give an example which shows the difference between the classes of mappings of bounded distortion and certain integrable distortion in terms of radial limits.     

Existence theorems for a second order nonlinear differential equation with nonlocal boundary conditions and their applications

By constructing an available integral operator and combining fixed point index theory with properties of Green’s function and Hölder’s inequality, this paper shows the existence of multiple positive solutions for a class of nonlocal boundary value problem of second-order differential equations λ x″(t)+w(t)f(t,x(t))=0, 0<t<1. The interesting point is that the term w(t) is L ^p -integrable for some 1≤p≤+∞. We illustrate our results by one example, which can not be handled using the existing results.     

On Mori's theorem in quasiconformal theory

In this paper, it is proved that the Mori constant, the Hölder coefficient of the familyQC _K (B) ofK-quasiconformal self-mappings of the unit diskB with the origin fixed, is at most 46^1?1 K. It is also shown that the Hölder coefficient of the restriction of a mapf∈QC _K (B) to the disk |z|≤ sin 25.7° is at most 16^1?1 K.     

Distortion theorems of plane quasiconformal mappings

The authors prove a conjecture on elliptic integrals and obtain sharp bounds for φK(r) and λ(K). By using Teichmüller's module theorem, the authors obtain a distortion theorem of K-quasiconformal mappings on the plane.     

Sharp bounds for the number of roots of univariate fewnomials

Let K be a field and t?0. Denote by Bm(t,K) the supremum of the number of roots in K^?, counted with multiplicities, that can have a non-zero polynomial in K[x] with at most t+1 monomial terms. We prove, using an unified approach based on Vandermonde determinants, that Bm(t,L)?t²Bm(t,K) for any local field L with a non-archimedean valuation v:L→R∪{∞} such that vZ≠0_|≡0 and residue field K, and that Bm(t,K)?(t²−t+1)(pf−1) for any finite extension K/Qp with residual class degree f and ramification index e, assuming that p>t+e. For any finite extension K/Qp, for p odd, we also show the lower bound Bm(t,K)?(2t−1)(pf−1), which gives the sharp estimation Bm(2,K)=3(pf−1) for trinomials when p>2+e.     

Infinite Multiplicity of Positive Entire Solutions for a Semilinear Elliptic Equation

We establish that the elliptic equation Δu+K(x)u^p+μf(x)=0 in Rⁿ has infinitely many positive entire solutions for small μ?0 under suitable conditions on K, p, and f.     

The Möbius midpoint condition as a test for quasiconformality and the quasimöbius property

The Möbius midpoint condition, introduced by Goldberg in 1974 as a criterion for the quasisymmetry of a mapping of the line onto itself and considered by Aseev and Kuzin in 1998 in the same role for the topological embeddings of the line into ? ⁿ , yields no information on the quasiconformality or quasisymmetry of a topological embedding of ? ^k into ? ⁿ for 1 < k ≤ n. In this article we introduce a Möbius-invariant modification of the midpoint condition, which we call the “Möbius midpoint condition” MMC(f) ≤ H < 1. We prove that if this condition is fulfilled then every homeomorphism of domains in \(\overline {\mathbb{R}^n }\) is K(H)-quasiconformal, while a topological embedding of the sphere \(\overline {\mathbb{R}^k }\) into \(\overline {\mathbb{R}^n }\) (for 1 ≤ k ≤ n) is ω _H-quasimöbius. The quasiconformality coefficient of K(H) and the distortion function ω _H depend only on H and are expressed by explicit formulas showing that K(H) → 1 and ω _H → id as H → 1/2. Since MMC(f) = 1/2 is equivalent to the Möbius property of f, the resulting formulas yield the closeness of the mapping to a Möbius mapping for H near 1/2.     

On the index of approximating sets of periodic points

Summary LetK be a compact space andf:K→K a continuous map without fixed points, i.e. Fixf=⊘. For prime numbersp, the sets Fixf ^p are freeℤ/p-spaces with theℤ/p-action induced byf. Our aim is to estimate the topological indicesi(F _p,f) of invariant subsetsF _p⊂Fixf ^p approximating a givenS⊂K. We construct an example (K,f,S) withS⊂Fixf ^q (q being some prime number) such that, for each neighborhoodU ofS, i (Fix (f|u) ^p, f) increases linearly withp. This article was processed by the author using the L^AT_EX style filecljour1 from Springer-Verlag.     

A Class of Nonlinear Averaging Integral Operators

LetHbe the class of analytic functions defined in the unit discU, and let coEdenote the convex hull of a setEinC. IfK⊂H, then an operatorI:K→His an averaging operator ifI[f](0) =f(0) andI[f](U) ⊂ cof(U), for allf∈K. The authors show that the operatorI_β,γ[f](z) ≡ [γz^−γ∫^z₀f^β(t)t^γ−1dt]^1/βis an averaging operator on certain subsets ofH.     

Composition operators on W 1 X are necessarily induced by quasiconformal mappings

Let Ω ? ? ⁿ be an open set and X(Ω) be any rearrangement invariant function space close to L ^q (Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ? u ? f from W ¹ X to W ¹ X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.