Recent zbMATH articles in MSC 30C62https://zbmath.org/atom/cc/30C622023-03-23T18:28:47.107421ZWerkzeugBernstein-Walsh-type inequalities for derivatives of algebraic polynomials on the regions of complex planehttps://zbmath.org/1503.300212023-03-23T18:28:47.107421Z"Özkartepe, Naciye Pelin"https://zbmath.org/authors/?q=ai:ozkartepe.naciye-pelin"Gün, Cevahir Doğanay"https://zbmath.org/authors/?q=ai:gun.cevahir-doganay"Abdullayev, Fahreddin"https://zbmath.org/authors/?q=ai:abdullayev.fahreddin-gSummary: 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.Quasiconformal maps with thin dilatationshttps://zbmath.org/1503.300502023-03-23T18:28:47.107421Z"Bishop, Christopher J."https://zbmath.org/authors/?q=ai:bishop.christopher-jThe author gives an estimate that quantifies the fact that a normalized quasiconformal map whose dilatation is non-zero only on a set of small area approximates the identity uniformly on the whole plane. To be more precise, a measurable set \(E\) in the complex plane \(\mathbb C\) is said to be \((\epsilon, h)\)-thin if \(\epsilon>0\) and area\((E\cap D(z, 1))\le\epsilon h(|z|)\) for all \(z\in\mathbb C\), where \(h:[0, \infty]\to [0, \pi]\) is a bounded decreasing function such that \(\int_0^{\infty}h(r)r^ndr<\infty\) for every \(n>1\). Let \(F\) be a quasiconformal map on the whole plane which is conformal near \(\infty\) and satisfies the normalized condition \(F(z)=z+O(1/|z|)\) as \(z\to\infty\). Suppose the complex dilatation \(\mu\) is small in the sense that \(E=\{z: \mu(z)\neq 0\}\) is \((\epsilon, h)\)-thin. Then for all \(z\in\mathbb C\) \[|F(z)-z|\le\frac{\epsilon^{\beta}}{|z|+1},\] where \(\beta>0\) depends only on \(\|\mu\|_{\infty}\) and \(h\). In particular, as \(\epsilon\to 0\), \(F\) converges uniformly to the identity on the whole plane.
Reviewer: Yuliang Shen (Suzhou)Dimension compression and expansion under homeomorphisms with exponentially integrable distortionhttps://zbmath.org/1503.300512023-03-23T18:28:47.107421Z"Hitruhin, Lauri"https://zbmath.org/authors/?q=ai:hitruhin.lauriSummary: We improve both dimension compression and expansion bounds for homeomorphisms with \(p\)-exponentially integrable distortion. To the first direction, we also introduce estimates for the compression multifractal spectra, which will be used to estimate compression of dimension, and for the rotational multifractal spectra. For establishing the expansion case, we use the multifractal spectra of the inverse mapping and construct examples proving sharpness.Integrable Teichmüller spacehttps://zbmath.org/1503.300522023-03-23T18:28:47.107421Z"Liu, Xueping"https://zbmath.org/authors/?q=ai:liu.xueping"Shen, Yuliang"https://zbmath.org/authors/?q=ai:shen.yuliangSummary: In this note we will consider to what extent some known results on the \(p\)-integrable Teichmüller space \(T_p\) for \(p\ge 2\) can be extended to the case \(1<p<2\), giving several new characterizations for \(p\)-integrable quasicircles. We will also introduce a new concept \(T^s_p\) which we call the strong \(p\)-integrable Teichmüller space and discuss various models of \(T^s_p\) by means of (analytic) Besov space theory.On the quasiconformal equivalence of dynamical Cantor setshttps://zbmath.org/1503.300532023-03-23T18:28:47.107421Z"Shiga, Hiroshige"https://zbmath.org/authors/?q=ai:shiga.hiroshigeSummary: The complement of a Cantor set in the complex plane is itself regarded as a Riemann surface of infinite type. The problem of this paper is the quasiconformal equivalence of such Riemann surfaces. Particularly, we are interested in Riemann surfaces given by Cantor sets which are created through dynamical methods. discuss the quasiconformal equivalence for the complements of Cantor Julia sets of rational functions and generalized Cantor sets. We also consider the Teichmüller distance between generalized Cantor sets.A real-variable construction with applications to BMO-Teichmüller theoryhttps://zbmath.org/1503.300542023-03-23T18:28:47.107421Z"Wei, Huaying"https://zbmath.org/authors/?q=ai:wei.huaying"Zinsmeister, Michel"https://zbmath.org/authors/?q=ai:zinsmeister.michelA strongly quasisymmetric homeomorphisms of the unit circle \(\mathbb{S}\) is an element of the BMOA-Teichmüller space studied by \textit{K. Astala} and the second author [Math. Ann. 289, No. 4, 613--625 (1991; Zbl 0896.30028)]. In other words, the set of strongly quasisymmetric homeomorphisms is the group of homeomorphisms \(h\) of the unit circle \(\mathbb{S}\) for which \(h\) is absolutely continuous and \(|h'|\) belongs to the class of \(A_{\infty}\)-weights introduced by \textit{B. Muckenhoupt} [Trans. Am. Math. Soc. 165, 207--226 (1972; Zbl 0236.26016 )]. A weight \(\omega\) on the unit circle \(\mathbb{S}\) is called doubling if there exists a constant \(C\) such that, for all adjacent intervals \(I, I^{*}\subset \mathbb{S}\) of the same length, \(C^{-1}\omega(I)\leq \omega(I^{*})\leq C\omega(I)\), where \(\omega(I)=\int_{I}\omega(z)|dz|\). \textit{C. Fefferman} and \textit{B. Muckenhoupt} [Proc. Am. Math. Soc. 45, 99--104 (1974; Zbl 0318.26010)] showed that \(A_{\infty}\)-weights are doubling, and they also provided an example of a function that satisfies the doubling condition but is not in \(A_{\infty}\). However it was not known yet whether or not, under the assumption that \(\omega\) is a doubling weight, the condition \(\log \omega \in \mathrm{BMO}(\mathbb{S})\) itself implies that \(\omega\in A_{\infty}\).
In this paper, the authors, with the use of real-variable techniques, provide a negative answer to this question. They construct a weight function \(\omega\) on the interval \([0,2\pi)\) which is doubling and such that \(\log \omega\) is a BMO function, but which is not a Muckenhoupt weight in\(A_{\infty}\). Applications to the BMO-Teichmüller spaces and the space of chord-arc curves are considered.
Reviewer: Shengjin Huo (Tianjin)Harmonic quasi-isometries of pinched Hadamard surfaces are injectivehttps://zbmath.org/1503.531292023-03-23T18:28:47.107421Z"Benoist, Yves"https://zbmath.org/authors/?q=ai:benoist.yves"Hulin, Dominique"https://zbmath.org/authors/?q=ai:hulin.dominiqueThe authors prove that a harmonic quasi-isometric map between pinched Hadamard surfaces is a quasi-conformal diffeomorphism.
Reviewer: Vladimir Balan (Bucureşti)