Chord-arc Curves Research Articles

Hölder classes in the 𝐿^{𝑝} norm on a chord-arc curve in ℝ³

The Hölder classes L p α ( L ) L_p^{\alpha } (L) in the L p ( L ) L_p(L) norm on a chord-arc curve L L in R 3 \mathbb {R}^3 are defined and direct and inverse approximation theorems are proved for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the L p ( L ) L^p(L) norm, the direct theorem is proved for a certain subclass of L p α ( L ) L^\alpha _p(L) and the inverse theorem covers the entire Hölder class.

Parametrization of the p-Weil–Petersson Curves: Holomorphic Dependence

Similar to the Bers simultaneous uniformization, the product of the p-Weil–Petersson Teichmüller spaces for p ge 1 provides the coordinates for the space of p-Weil–Petersson embeddings gamma of the real line {mathbb {R}} into the complex plane {mathbb {C}}. We prove the biholomorphic correspondence from this space to the p-Besov space of u=log gamma ' on {mathbb {R}} for p>1. From this fundamental result, several consequences follow immediately which clarify the analytic structures concerning parameter spaces of p-Weil–Petersson curves. Specifically, it implies that the correspondence of the Riemann mapping parameters to the arc-length parameters preserving the images of curves is a homeomorphism with bi-real-analytic dependence of the change of parameters. This is analogous to the classical theorem of Coifman and Meyer for chord-arc curves.

BMO embeddings, chord-arc curves, and Riemann mapping parametrization

We consider the space of chord-arc curves on the plane passing through infinity with their parametrization γ defined on the real line, and embed this space into the product of the BMO Teichmüller spaces. The fundamental theorem we prove about this representation is that γ↦log⁡γ′ is a biholomorphic homeomorphism into the complex Banach space of BMO functions. Using these two equivalent complex structures, we develop a clear exposition on the analytic dependence of involved mappings between certain subspaces. Especially, we examine the parametrization of a chord-arc curve by using the Riemann mapping and its dependence on the arc-length parametrization. As a consequence, we can solve the conjecture of Katznelson, Nag, and Sullivan by showing that this dependence is not continuous.

Lp-norm approximation of Holder functions by harmonic functions on some multidimensional compact sets

In this paper we consider the class of H¨older functions in the sense of Lp norm on certain compacts in Rm (m >= 3) and prove theorems on approximation by functions harmonic in neighborhoods of these compacrs. These compacts are a generalization to the higher dimensions of the concept of chord-arc curve in R3. The size of the neighborhood decreases along with an increase in the accuracy of the approximation. Estimates of the approximation rate and the gradient of the approximation functions are made in the same Lp-norm.

On the BMO boundedness of the Cauchy integral on a chord-arc curve

On the BMO boundedness of the Cauchy integral on a chord-arc curve

Space of chord-arc curves and BMO/VMO Teichmüller space

This paper focuses on the structure of the subspace \(T_c\) of the BMO Teichmüller space \(T_b\) corresponding to chord-arc curves, which contains the VMO Teichmüller space \(T_v\). We prove that \(T_c\) is not a subgroup with respect to the group structure of \(T_b\), but it is preserved under the inverse operation and the left and the right translations by any element of \(T_v\). Moreover, we show that \(T_b\) has a fiber structure induced by \(T_v\), and the complex structure of \(T_b\) can be projected down to the quotient space \(T_v \backslash T_b\). Then, we see that \(T_c\) consists of fibers of this projection, and its quotient space also has the induced complex structure.

A real-variable construction with applications to BMO–Teichmüller theory

With the use of real-variable techniques, we construct a weight function ω on the interval [0,2π) which is doubling and satisfies logω is a BMO function, but which is not a Muckenhoupt weight (A∞). Applications to the BMO–Teichmüller space and the space of chord-arc curves are considered.

The Two-Dimensional Liquid Crystal Droplet Problem with a Tangential Boundary Condition

This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume constraint. We establish in 2D the existence of an optimal shape that has two cusps on the boundary. We also prove that the boundary of the droplet is a chord–arc curve with its normal vector field in the VMO space, and its arc-length parameterization belongs to the Sobolev space $$H^{3/2}$$ . In fact, the boundary curves of such droplets closely resemble the so-called Weil–Petersson class of planar curves. In addition, the asymptotic behavior of the optimal shape when the volume becomes extremely large or small is studied.

Constructive description of Hölder classes on some multidimensional compact sets

We give a constructive description of Hölder classes of functions on certain compacts in Rm (m > 3) in terms of a rate of approximation by harmonic functions in shrinking neighborhoods of these compacts. The considered compacts are a generalization to the higher dimensions of compacts that are subsets of a chord-arc curve in R3. The size of the neighborhood is directly related to the rate of approximation it shrinks when the approximation becomes more accurate. In addition to being harmonic in the neighborhood of the compact the approximation functions have a property that looks similar to Hölder condition. It consists in the fact that the difference in values at two points is estimated in terms of the size of the neighborhood, if the distance between these points is commensurate with the size of the neighborhood (and therefore it is estimated in terms of the distance between the points).

Constructive description of Hölder-like classes on an arc in [formula omitted] by means of harmonic functions

We give a constructive description of Hölder-like classes of functions on chord-arc curves in R3 in terms of a rate of approximation by harmonic functions in shrinking neighborhoods of those curve.

Ahlfors-regular curves and Carleson measures

We study the relation between the boundary of a simply connected domain $\Omega$ being Ahlfors-regular and the invariance of Carleson measures under the push-forward operator induced by a conformal mapping from the unit disk $\Delta$ onto the domain $\Omega$. As an application, we characterize the chord-arc curves with small norms and the asymptotically smooth curves in terms of the complex dilatation of some quasiconformal reflection with respect to the curve.

Cauchy independent measures and almost-additivity of analytic capacity

We show that, given a family of discs centered at a chord-arc curve, the analytic capacity of a union of subsets of these discs (one subset in each disc) is comparable with the sum of their analytic capacities. However, we need the discs in question to be separated, and it is not clear whether the separation condition is essential. We apply this result to find families {μj} of measures in ℂ with the following property. If the Cauchy integral operators $${C_{{\mu _j}}}$$ from L2(μj) to itself are bounded uniformly in j, then Cμ, μ = Σμj, is also bounded from L2(μ) to itself.

Carleson measures and chord-arc curves

Following Semmes and Zinsmeister, we continue the study of Carleson measures and their invariance under pull-back and push-forward operators. We also study the analogous statements for vanishing Carleson measures. As an application, we show that some quotient space of the space of chord-arc curves has a natural complex structure.

POLYGONAL QUASICONFORMAL MAPPINGS AND CHORD-ARC CURVES

In this paper we show that a polygonal quasiconformal mapping always corresponds to a chord-arc curve. Furthermore, we find that the set of curves corresponding to polygonal quasiconformal mappings is path connected in the set of all bounded chord-arc curves.

Weak Chord-Arc Curves and Double-Dome Quasisymmetric Spheres

Abstract Let Ω be a planar Jordan domain and α > 0. We consider double-dome-like surfaces Σ(Ω, tα) over Ω where the height of the surface over any point x ∈ Ωequals dist(x, ∂Ω)α. We identify the necessary and sufficient conditions in terms of and α so that these surfaces are quasisymmetric to S2 and we show that Σ(Ω, tα) is quasisymmetric to the unit sphere S2 if and only if it is linearly locally connected and Ahlfors 2-regular.

Dirichlet problem and Sokhotski-Plemelj jump formula on Weil-Petersson class quasidisks

We show the solvability of the Dirichlet problem on Weil–Petersson class quasidisks and establish a Sokhotski–Plemelj jump formula for Weil–Petersson class quasicircles. Furthermore we show that the resulting Cauchy projections are bounded. In both cases the boundary data belongs to a certain conformally invariant Besov space. Moreover we show that the WP-class quasicircles are chord-arc curves.

Chord-arc curves and the Beurling transform

We study the relation between the geometric properties of a quasicircle~$\Gamma$ and the complex dilatation~$\mu$ of a quasiconformal mapping that maps the real line onto~$\Gamma$. Denoting by~$S$ the Beurling transform, we characterize Bishop-Jones quasicircles in terms of the boundedness of the operator~$(I-\mu S)$ on a particular weighted $L^2$~space, and chord-arc curves in terms of its invertibility. As an application we recover the~$L^2$ boundedness of the Cauchy integral on chord-arc curves.

Sets of constant distance from a Jordan curve

We study the $\epsilon$-level sets of the signed distance function to a planar Jordan curve $\Gamma$, and ask what properties of $\Gamma$ ensure that the $\epsilon$-level sets are Jordan curves, or uniform quasicircles, or uniform chord-arc curves for all sufficiently small $\epsilon$. Sufficient conditions are given in term of a scaled invariant parameter for measuring the local deviation of subarcs from their chords. The chordal conditions given are sharp.

Location of geodesics and isoperimetric inequalities in Denjoy domains

AbstractWe find approximate solutions (chord–arc curves) for the system of equations of geodesics in Ω∩ℍ for every Denjoy domain Ω, with respect to both the Poincaré and the quasi-hyperbolic metrics. We also prove that these chord–arc curves are uniformly close to the geodesics. As an application of these results, we obtain good estimates for the lengths of simple closed geodesics in any Denjoy domain, and we improve the characterization in a 1999 work by Alvarez et al. on Denjoy domains satisfying the linear isoperimetric inequality.

Bilipschitz homogeneous Jordan curves, Möbius maps, and dimension

We characterize fractal chordarc curves in Euclidean space by the fact that they remain bilipschitz homogeneous under inversion. We illustrate this result by constructing two examples. The techniques used in these constructions provide a means of calculating various dimensions of bilipschitz homogeneous Jordan curves.