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Michel Zinsmeister

Publications and source records attributed to Michel Zinsmeister.

At least 19 recordsLinked to original sources
arXiv2026

p-Dirichlet spaces over chord-arc domains

Let $Γ$ be a rectifiable Jordan curve in the complex plane, let $Ω_i$ and $Ω_e$ be its interior and exterior domains, respectively, and let $1 < p < \infty$. Let $E$ be the vector space of restrictions to $Γ$ of functions in $C^1(\mathbb C)$. We consider the following three seminorms on $E$: (i) $\lVert u\rVert_i=\left(\frac{1}{2π}\iint_{Ω_i}|\nabla U_i(z)|^pλ_{Ω_i}^{2-p}(z)\,dA(z)\right)^{1/p}$, where $U_i$ is the harmonic extension of $u$ to $Ω_i$ and $λ_{Ω_i}$ is the hyperbolic density of $Ω_i$; (ii) $\lVert u\rVert_e$, defined analogously on $Ω_e$; and (iii) $\lVert u\rVert_{B_p(Γ)}=\left(\frac{1}{4π^2}\iint_{Γ\timesΓ}\frac{|u(z)-u(ζ)|^p}{|z-ζ|^2}\,|dz|\,|dζ|\right)^{1/p}$. These three seminorms are known to be equivalent when $Γ$ is a chord-arc curve. We investigate the converse problem.

math.CV↗
arXiv2026

Fractional Besov-Sobolev Spaces on Quasicircles

Let $Γ$ be a bounded Jordan curve and $Ω_i,Ω_e$ its two complementary components. For $p\in (1, \infty),\,s\in(0,1)$ we define the two spaces $\mathcal{B}_{p,p}^s(Ω_{i,e})$ as the set of harmonic functions $u$ respectively in $Ω_i$ and $Ω_e$ such that $$ \iint_{Ω_{i,e}} |\nabla u(z)|^p d(z,Γ)^{(1-s)p-1} dxdy<+\infty.$$ When it is possible to identify these spaces with spaces of functions on the boundary (trace spaces), we address the question of their equality. When $Γ$ is the unit circle, these two spaces coincide with homogeneous fractional Besov-Sobolev spaces and the framework of quasicircles appears to be an appropriate generalization. In this framework, we study the boundedness of the Plemelj-Calderón operator and apply the results to show that for some values of $p,s$, if the two spaces coincide, they are restrictions to $Γ$ of some weighted Sobolev space. If $Γ$ is further assumed to be rectifiable, we define $B_{p,p}^s(Γ)$ as the space of functions $f\in L^p(Γ)$ such that $$\iint_{Γ\times Γ}\frac{|f(z)-f(ζ)|^p}{|z-ζ|^{1+ps}} |dz||dζ|<+\infty.$$ Again, these spaces coincide with the homogeneous fractional Besov-Sobolev spaces for the unit circle. While the chord-arc property is the necessary and sufficient condition for the equality $$\mathcal{B}_{p,p}^s(Ω_{i})=\mathcal{B}_{p,p}^s(Ω_{e})=B_{p,p}^s(Γ)$$ in the case of $s=1/p,\, p\ge 2$, this is no longer the case for general $s\in (0,1)$. However, we show that equality holds for radial-Lipschitz curves. Finally, we re-interpretate some of our results as some "almost"-Dirichlet principle in the spirit of Maz'ya.

math.CV↗
arXiv2025

Cauchy Integral, Fractional Sobolev Spaces and Chord-Arc Curves

Let $Γ$ be a bounded Jordan curve and $Ω_i,Ω_e$ its two complementary components. For $s\in(0,1)$ we define $\mathcal{H}^s(Γ)$ as the set of functions $f:Γ\to \mathbb C$ having harmonic extension $u$ in $Ω_i\cup Ω_e$ such that $$ \iint_{Ω_i\cup Ω_e} |\nabla u(z)|^2 d(z,Γ)^{1-2s} dxdy<+\infty.$$ If $Γ$ is further assumed to be rectifiable we define $H^s(Γ)$ as the space of measurable functions $f:Γ\to \mathbb C$ such that $$\iint_{Γ\times Γ}\frac{|f(z)-f(ζ)|^2}{|z-ζ|^{1+2s}} dσ(z)dσ(ζ)<+\infty.$$ When $Γ$ is the unit circle these two spaces coincide with the homogeneous fractional Sobolev space defined via Fourier series. For a general rectifiable curve these two spaces need not coincide and our first goal is to investigate the cases of equality: while the chord-arc property is the necessary and sufficient condition for equality in the classical case of $s=1/2$, this is no longer the case for general $s\in (0,1)$. We show however that equality holds for Lipschitz curves. The second goal involves the Plemelj-Calderón problem. ......

math.CV↗
arXiv2024

Dirichlet spaces over chord-arc domains

If $U$ is a $C^{\infty}$ function with compact support in the plane, we let $u$ be its restriction to the unit circle $\mathbb{S}$, and denote by $U_i,\,U_e$ the harmonic extensions of $u$ respectively in the interior and the exterior of $\mathbb S$ on the Riemann sphere. About a hundred years ago, Douglas has shown that \begin{align*} \iint_{\mathbb{D}}|\nabla U_i|^2(z)dxdy&= \iint_{\bar{\mathbb{C}}\backslash\bar{\mathbb{D}}}|\nabla U_e|^2(z)dxdy &= \frac{1}{2π}\iint_{\mathbb S\times\mathbb S}\left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|^2|dz_1||dz_2|, \end{align*} thus giving three ways to express the Dirichlet norm of $u$. On a rectifiable Jordan curve $Γ$ we have obvious analogues of these three expressions, which will of course not be equal in general. The main goal of this paper is to show that these $3$ (semi-)norms are equivalent if and only if $Γ$ is a chord-arc curve.

math.CV↗
arXiv2023

Denjoy Domains and BMOA

A Denjoy domain is a plane domain whose complement is a closed subset $E$ of the extended real line $\bar{R}$ containing $\infty$ : such a domain is called Carleson-homogeneous if there exists $C>0$ such that for all $z\in E$ and $r>0$, one has $\vert E\cap [z-r,z+r]\vert\geq Cr$, where $\vert\cdot\vert$ is the Lebesgue measure on the line. We prove that if $U=\bar{ \mathbb C}\backslash K$ is a Carleson-homogeneous Denjoy domain then, if $f$ stands for one of its universal coverings, $\log {f'}\in BMOA.$ In order to prove this result, we develop ideas from [On Carleson measures induced by Beltrami coefficients being compatible with Fuchsian groups, Ann. Fenn. Math. 46(2021),67-77] leading to a general theorem about planar domains giving sufficient conditions ensuring that $\log {f'}\in BMOA$ for any universal covering $f.$

math.CV↗
arXiv2023

Complex Generalized Integral Means Spectrum of Drifted Whole-Plane SLE and LLE

We present new results for the complex generalized integral means spectrum for two kinds of whole-plane Loewner evolutions driven by Lévy processes: - Lévy processes with continuous trajectories, which correspond to Schramm-Loewner evolutions (SLE) with a drift term in the Brownian driving function. A natural path to access the standard integral means spectrum in the presence of drift goes through the introduction of the complex generalized integral means spectrum, which is obtained via the so-called Liouville quantum gravity. -Symmetric Lévy processes for which we generalize recent results by Loutsenko and Yermolayeva.

math-ph↗
arXiv2022

A new Model of City Growth and its Application to a middle sized French City

In the first part of this paper we propose a new theoretical model of city growth based on percolation. The second half oh the paper is devoted to a concrete application of the model, namely to the city of Montargis. It appears that the embedded algorithm is quite efficient in terms of computational time and allows to exploit big data type ressources such as individual land lots.

physics.soc-ph↗
arXiv2021

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 $[0, 2π)$ that is doubling and satisfies $\log ω$ is a BMO function, but which is not a Muckenhoupt weight ($A_\infty$). Applications to the BMO-Teichmüller space and the space of chord-arc curves are considered.

math.CV↗
arXiv2019

On Ruelle's property

In this paper we investigate the range of validity of Ruelle's property. First, we show that every finitely-generated Fuchsian group has Ruelle's property. We also prove the existence of an infinitely-generated Fuchsian group satisfying Ruelle's property. Concerning the negative results we first generalize Astala-Zinsmeister's results by proving that all convergence Fuchsian groups of the first kind fail to have Ruelle's property. At last, we also give some results about the second kind Fuchsian groups.

math.CV↗
arXiv2019

Local Analysis of Loewner Equation

Let $λ:[0,+\infty)\mapsto\mathbb{R}$ be the driving function of a chordal Loewner process. In this paper we find new conditions on $λ$ which imply that the process is generated by a simple curve. This result improves former one by Lind ,Marshall and Rhode, and it particular gives new results about the case $λ(t)=cW_b(t)$, $W_b$ being a Hölder-$1/2$ Weierstrass function. In the second part we find new conditions on $λ$ implying that the process is generated by a curve. The main tool here is a duality relation between the real part and the imaginary part of the Loewner equation.

math.CV↗
arXiv2018

Ahlfors-regular curves and Carleson measures

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

math.CV↗
arXiv2017

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.

math.CV↗
arXiv2017

BMO-Teichmüller spaces revisited

In a paper of Cui and Zinsmeister the equivalence among three definitions of BMO-Teichmüller spaces associated with a Fuchsian group was proven using the Douady-Earle extension operator. In this paper, we show that these equivalences are actually biholomorphisms. It was further shown in the above quoted paper that the Douady-Earle extension operator is continuous at the origin. We improve this result by showing Gâteaux-differentiability at this point.

math.CV↗
arXiv2017

On The Brownian Loop Measure

In 2003 Lawler and Werner introduced the Brownian loop measure and studied some of its properties. Cardy and Gamsa has predicted a formula for the total mass of the Brownian loop measure on the set of simple loops in the upper half plane and disconnect two given points from the boundary. In this paper we give a rigorous proof of the formula using a result by Beliaev and Viklund and heavy computations.

math-ph↗
arXiv2017

Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

We consider the whole-plane SLE conformal map f from the unit disk to the slit plane, and show that its mixed moments, involving a power p of the derivative modulus |f'| and a power q of the map |f| itself, have closed forms along some integrability curves in the (p,q) moment plane, which depend continuously on the SLE parameter kappa. The generalization of this integrability property to the m-fold transform of f is also given. We define a generalized integral means spectrum corresponding to the singular behavior of the mixed moments above. By inversion, it allows for a unified description of the unbounded interior and bounded exterior versions of whole-plane SLE, and of their m-fold generalizations. The average generalized spectrum of whole-plane SLE takes four possible forms, separated by five phase transition lines in the moment plane, whereas the average generalized spectrum of the m-fold whole-plane SLE is directly obtained from a linear map acting in that plane. We also conjecture the form of the universal generalized integral means spectrum.

math-ph↗
arXiv2017

Integral means spectrum of whole-plane SLE

We complete the mathematical analysis of the fine structure of harmonic measure on SLE curves that was initiated by Beliaev and Smirnov, as described by the averaged integral means spectrum. For the unbounded version of whole-plane SLE as studied by Duplantier, Nguyen, Nguyen and Zinsmeister, and Loutsenko and Yermolayeva, a phase transition has been shown to occur for high enough moments from the bulk spectrum towards a novel spectrum related to the point at infinity. For the bounded version of whole-plane SLE studied here, a similar transition phenomenon, now associated with the SLE origin, is proved to exist for low enough moments, but we show that it is superseded by the earlier occurrence of the transition to the SLE tip spectrum.

math-ph↗
arXiv2014

The Coefficient Problem and Multifractality of Whole-Plane SLE and LLE

We revisit the Bieberbach conjecture in the framework of SLE processes and, more generally, Lévy processes. The study of their unbounded whole-plane versions leads to a discrete series of exact results for the expectations of coefficients and their variances, and, more generally, for the derivative moments of some prescribed order p. These results are generalized to the m-fold conformal maps of whole-plane SLEs or Lévy-Loewner Evolutions (LLEs). We also study the (averaged) integral means multifractal spectra of these unbounded whole-plane SLE curves. We prove the existence of a phase transition at a certain moment order, at which one goes from the bulk SLE expected integral means spectrum, as established by Beliaev and Smirnov, to a new integral means spectrum. The latter is furthermore shown to be intimately related, via the associated packing spectrum, to radial SLE derivative exponents, and to local SLE tip multifractal exponents obtained from quantum gravity. This is generalized to the integral means spectrum of the m-fold transform of the unbounded whole-plane SLE map.

math-ph↗