Shift coordinates,stretch lines and polyhedral structures for Teichmüller space

This paper has two parts. In the first part, we study shift coordinates on a sphere S equipped with three distinguished points and a triangulation whose vertices are the distinguished points. These coordinates parametrize a space that we call an unfolded Teichmüller space. This space contains Teichmüller spaces of the sphere with boundary components and cusps (which we call generalized pairs of pants), for all possible values of and satisfying . The parametrization of by shift coordinates equips this space with a natural polyhedral structure, which we describe more precisely as a cone over an octahedron in . Each cone over a simplex of this octahedron is interpreted as a Teichmüller space of the sphere with boundary components and cusps, for fixed and , the sphere being furthermore equipped with an orientation on each boundary component. There is a natural linear action of a finite group on whose quotient is an augmented Teichmüller space in the usual sense. We describe several aspects of the geometry of the space . Stretch lines and earthquakes can be defined on this space. In the second part of the paper, we use the shift coordinates to obtain estimates on the behaviour of stretch lines in the Teichmüller space of a surface obtained by gluing hyperbolic pairs of pants. We also use the shift coordinates to give formulae that express stretch lines in terms of Fenchel-Nielsen coordinates. We deduce the disjointness of some stretch lines in Teichmüller space. We study in more detail the case of a closed surface of genus 2. Authors’ addresses: A. Papadopoulos, Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany; G. Théret, Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France and Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C, Denmark     

Global Representatives of Colombeau Holomorphic Generalized Functions

It is proved that a holomorphic generalized function in the sense of Colombeau has a representative consisting of a net of holomorphic functions. More generally, the existence of global representatives of Colombeau solutions to elliptic partial differential equations is proved. Further, it is shown that a generalized holomorphic function on Ω, which is equal to zero in an open set , is equal to zero. First author partially supported by FWF (Austria), grants P16820 and Y237. Third author partially supported by MNZŽ (Serbia), project 144016.     

Sums of the error term function in the mean square for ζ(s)

Sums of the form are investigated, where is the error term in the mean square formula for . The emphasis is on the case k = 1, which is more difficult than the corresponding sum for the divisor problem. The analysis requires bounds for the irrationality measure of e^2πm and for the partial quotients in its continued fraction expansion. Authors’ addresses: Y. Bugeaud, Université Louis Pasteur, Mathématiques, 7 rue René Descartes, F-67084 Strasbourg cedex, France; A. Ivić, Katedra Matematike RGF-a, Universitet u Beogradu, Đušina 7, 11000 Beograd, Serbia     

Faisceaux\bar{\partial}-cohérents sur les variétés complexes

Résumé Dans ce travail nous généralisons au contexte des faisceaux analytiques cohérents un résultat classique de Koszul–Malgrange (Koszul and Malgrange in Arch Math 9 : 102–109, 1958) concernant l’intégrabilité des connexions de type (0,1) sur un fibré vectoriel complexe C ^∞ au-dessus d’une variété complexe. En introduisant la notion de faisceau -cohérent, qui est une notion qui vit dans le contexte C ^∞, nous montrons l’existence d’une équivalence (exacte) entre la catégorie des faisceaux analytiques cohérents et la catégorie des faisceaux -cohérents. Une application de notre caractérisation est une méthode (la -stabilité) qui permet de trouver des structures analytiques par déformation C ^∞ d’autres structures analytiques.     

Fourier transformation of Sato's hyperfunctions

A new generalized function space in which all Gelfand-Shilov classes (α>1) of analytic functionals are embedded is introduced. This space of ultrafunctionals does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on R^k is constructed that extends the Fourier transformation of Roumieu-type ultradistributions and is naturally interpreted as the Fourier transformation of hyperfunctions. The notion of carrier cone that replaces the notion of support of a generalized function for ultrafunctionals is proposed. A Paley-Wiener-Schwartz-type theorem describing the Laplace transformation of ultrafunctionals carried by proper convex closed cones is obtained and the connection between the Laplace and Fourier transformations is established.     

Jean-Paul Penot

Discussion

Jean-Paul Penot Université de Pau et des Pays de l’Adour, France     

The structure of spaces of quasianalytic functions of Roumieu type

It is shown that spaces of quasianalytic ultradifferentiable functions of Roumieu type ℰ_{w}(Ω), on an open convex set , satisfy some new (Ω) -type linear topological invariants. Some consequences for the splitting of short exact sequences of these spaces as well as for the structure of the spaces are derived. In particular, Fréchet quotients of ℰ_{w}(Ω) have property (), while dual Fréchet quotients have property () of Vogt. The work of P. Domański was supported by Committee of Scientific Research (KBN), Poland, grant P03A 022 25.     

Problèmes d'extension dans les domaines convexes de type fini

Résumé Soient donnés D un domaine borné de , convexe, de type fini, X une hypersurface complexe telle que soit non vide, connexe et transverse et . Nous nous intéressons au problème suivant. Sous quelle(s) condition(s) existe-t-il telle que Φ|X = φ. Nous allons donner une condition nécessaire à l'existence de l'extension Φ puis sous une condition un peu plus forte nous montrerons la continuité d'un opérateur d'extension de type Berndtsson-Andersson.      

New spaces of functions and hyperfunctions for Hankel transforms and convolutions

In this paper we study the Hankel transformation and convolution on certain spaces of entire functions and its dual that is a space of hyperfunctions and contains the (even)-Schwartz space S _e ′. We prove that the Hankel transform is an automorphism of . Also the Hankel convolutors of are investigated. Authors’ addresses: Jorge J. Betancor, Claudio Jerez and Lourdes Rodríguez-Mesa, Departamento de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n, 38271 La Laguna (Sta. Cruz de Tenerife), Espa?a; Sandra M. Molina, Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350 (7600), Mar del Plata, Argentina     

From Brunn–Minkowski to sharp Sobolev inequalities

We present a simple direct proof of the classical Sobolev inequality in with best constant from the geometric Brunn–Minkowski–Lusternik inequality. Research supported in part by NSF Gr. No. 0405587 and B. Zegarlinski’s Pierre de Fermat Grant 2006 from the Région Midi-Pyrénées, France.     

Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface {x ⁰ + x ⁿ = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation.     

Analytic Multifunctions, Holomorphic Motions and Hausdorff Dimension in IFSs

We study iterated function systems of contractions which depend holomorphically on a complex parameter λ. We first restrict our attention to systems which consist of similarities that satisfy the OSC. In this setting, we prove that the Hausdorff dimension of the limit set J(λ) is a continuous, subharmonic function of λ. In the remainder of the paper, systems consisting of conformal contractions are considered. We give conditions under which J(λ) and A(λ) = describe a holomorphic motion, and construct an example that shows that this is not the case in general. We finally show that A(λ) is best described as an analytic multifunction of λ, a notion that generalizes that of holomorphic motion. This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Fonds Québécois de Recherche sur la Nature et les Technologies (FQRNT). This research was supported by the FQRNT.     

Dilatation estimates for quasiconformal extensions

LetD andD′ be ring domains inB ⁿ , withS ⁿ⁻¹ as one boundary component, and let be a homeomorphism which isK-quasiconformal inD and withf(S ⁿ⁻¹)=S ⁿ⁻¹. According to a result of Gehringf÷S ⁿ⁻¹ admits an extension which is quasiconformal inB ⁿ . We find here an upper bound for the dilatation ofg in terms ofn, K, and modD. This work was started during a visit to Université de Paris, financed by a cultural exchange program between France and Finland.     

Zeros of Generalized Holomorphic Functions

Let f be a generalized holomorphic function on a connected open set . It is proved that f equals zero if and only if there exists a smooth curve and a set A of positive (one-dimensional) measure such that f takes zero value on A. Also, a holomorphic generalized function different from zero on the disc, which takes zero values on a dense G _δ-set of the disc, is constructed. The generalized zero set of a holomorphic function is introduced and studied in an analogous way.     

<Emphasis Type="Italic">F</Emphasis>-isocristaux surconvergents et surcohérence différentielle

Résumé Soient un anneau de valuation discrète complet d’inégales caractéristiques, de corps résiduel parfait k, un -schéma formel propre et lisse, T un diviseur de la fibre spéciale P de , U l’ouvert de P complémentaire de T, Y un sous-k-schéma fermé lisse de U. Nous prouvons que la catégorie des F-isocristaux surconvergents sur Y est équivalente à celle des F-isocristaux surcohérents sur Y (voir [Car, 6.2.1 et 6.4.3.a)]). Plus généralement, nous établissons par recollement une telle équivalence pour tout k-schéma séparé lisse Y. Nous vérifions de plus que les F-complexes de -modules à cohomologie bornée et -surcohérente se dévissent en F-isocristaux surconvergents.     

Nonlinear Parabolic Equations with Regularized Derivatives in Colombeau Algebra

We consider several types of nonlinear parabolic equations with singular like potential and initial data. To prove the existence-uniqueness theorems we employ regularized derivatives. As a framework we use Colombeau space and Colombeau vector space      

Bounded tightness for weak topologies

In every Hausdorff locally convex space for which there exists a strictly finer topology than its weak topology but with the same bounded sets (like for instance, all infinite dimensional Banach spaces, the space of distributions or the space of analytic functions in an open set , etc.) there is a set A such that 0 is in the weak closure of A but 0 is not in the weak closure of any bounded subset B of A. A consequence of this is that a Banach space X is finite dimensional if, and only if, the following property [P] holds: for each set and each x in the weak closure of A there is a bounded set such that x belongs to the weak closure of B. More generally, a complete locally convex space X satisfies property [P] if, and only if, either X is finite dimensional or linearly topologically isomorphic to . Received: 11 June 2003     

On some bilinear problems on weighted Hardy‐Sobolev spaces

In this paper we study the bilinear problem of characterizing the positive Borel measures μ on S ⁿ, satisfying where $H_s^2(w)$ and $H_t^2(w)$ are weighted Hardy‐Sobolev spaces, under adequate conditions on the weight w.     

The property (DN) and the exponential representation of holomorphic functions

It is shown that a nuclear Fréchet spaceE has the property (DN) if and only if every holomorphic function onE ^*, the strongly dual space ofE, with values in the strongly dual space of a Fréchet spaceF having the property ( ) can be represented in the exponential form. Moreover, it is shown that the space of holomorphic functions onC , the space of all complex number sequences, has a linearly absolutely exponential representation system. But the space of holomorphic functions onE ^* does not have such a system whenE is a nuclear Fréchet space that does not have the property (DN).Supported by the State Program for Fundamental Researches in Natural Sciences     

Invariants and spaces of zero solutions of linear partial differential operators

It is shown that for open convex , d > 1 and a nontrivial polynomial P the space does not have property . If P is elliptic or homogeneous, then this holds for every open Ω. For even cannot occur and if it occurs for some Ω, then P must be hypoelliptic. Received: 18 July 2005