Showing papers in "Journal D Analyse Mathematique in 1991"

A vector-valued approach to tent spaces

Abstract: In this paper we present a new point of view to study the tent spaces introduced by Coifman, Meyer and Stein ([CMS1] and [CMS2]) by immersing them into vector-valued Lebesgue, Hardy and BMO spaces. This approach allows us to derive many of the known properties for tent spaces in a very simple manner. In fact most of the results are obtained as a consequence of similar results for those vector-valued spaces where they are immersed. The main tool we use to obtain such immersions and the applications to boundedness of operators, given in § 4, is the vector-valued Calderon-Zygmund theory.

Uniform, sobolev extension and quasiconformal circle domains

Abstract: This paper contributes to the theory of uniform domains and Sobolev extension domains. We present new features of these domains and exhibit numerous relations among them. We examine two types of Sobolev extension domains, demonstrate their equivalence for bounded domains and generalize known sufficient geometric conditions for them. We observe that in the plane essentially all of these domains possess the trait that there is a quasiconformal self-homeomorphism of the extended plane which maps a given domain conformally onto a circle domain. We establish a geometric condition enjoyed by these plane domains which characterizes them among all quasicircle domains having no large and no small boundary components.

A canonical structure theorem for finite joining-rank maps

Abstract: A numerical isomorphism invariant,joining-rank, was introduced in [1] as a quantitative generalization of Rudolph’s property of minimal self-joinings. Therein, a structure theory was developed for those transformationsT whose joining-rank, jr(T), is finite. Here, we sharpen the theorem and show it to be canonical: If jr(T) wheree andp are natural numbers andS is a map with minimal self-joinings such thatT is ane-point extension ofS p. Furthermore, the producte.p equals the joining-rank ofT. This theorem applies to any finite-rank mixing map, since for such maps the rank dominates the joining-rank. Another corollary is that any rank-1, transformation which is partial-mixing has minimal self-joinings. This partially answers a question of [3].

Lipschitz conditions in conformally invariant metrics

Abstract: We consider two conformally invariant metrics in proper subdomains of euclideann-spaceR n. We show that Lipschitz mappings in these metrics include the class of quasiconformational mappings as a proper subclass, yet these Lipschitz mappings have many properties similar to those of quasiconformal mappings.

A Morera type theorem forL2 functions in the Heisenberg group

Abstract: We obtain a Morera type characterization for the square integrable CR functions on the Heisenberg group.

Circle packing and riemann surfaces

Abstract: In Theorem 1 and Proposition 2 we generalize these results of Thurston to com­ pact bordered triangulated surfaces. Our proof relies on Thurston's result for closed surfaces. It follows the lines of Perron's method for solving the Dirichlet Problem using subharmonic functions. In Perron's method, the harmonic solu­ tion is the upper envelope of a family of subharmonic functions. To actually pro­ duce the harmonic solution one must exhibit suitable subharmonic functions (barriers) in the Perron family. The circle packing case is analogous: we obtain the solution as the upper envelope of a Perron-type family. Nonsingular circle packings are the analogues of harmonic functions, and circle packings with neg­ ative curvature cone type singularities are the analogues of subharmonic functions. We use Thurston's result for closed surfaces in order to produce a negative cur­ vature circle packing and thereby show that the family is nontrivial. The bound­ ary value problem and its solution by Perron family techniques were introduced by Peter Doyle [8] and Carter and Rodin [6,7]. They treated the euclidean case for genus :51; Beardon-Stephenson [4] treated the hyperbolic simply connected case by a related method. For the results above we need a suitable notion of "circle packing on a bordered surface" when circles on the border are to have infinite hyperbolic radius. There are two obvious possibilities for such a notion: infinite border circles can be re­ quired to be tangent to the border or they can be required to be orthogonal to the border. That is, a hyperbolic circle of infinite radius might be defined to be ei­ ther a horocycle or a hyperbolic geodesic. With either choice one can define the

Fonctions auxiliaires et fonctionnelles analytiques (I)

Abstract: Les progres recents en theorie des nombres transcendants sont venus principalement du development de methodes algebriques (lemmes de zeros, criteres d'independance algebrique). Un des principaux obstacles pour resoudre les conjectures encore ouvertes semble etre la faiblesse des arguments analytiques. Le but de ce travail est de developper systematiquement les constructions auxiliaires qui interviennent generalement dans le premier pas des demonstrations de transcendance. Dans cette premiere partie, nous utilisons le principle des tiroirs (lemme de Thue-Siegel) pour construire des fonctions auxiliaries en une ou plusieurs variables. Dans la, seconde, nous presenterons une construction duale, qui produit des fonctionnelles auxiliaires. Des consequences arithmetiques feront l’objet d’un autre texte.

Unbounded quadrature domains inR n (n≥3)

Abstract: As a result of the powerful tools of complex analysis a lot of problems have been solved in the theory of Q.D.s (quadrature domain) inR 2. These problems are almost untouched inR n (n≥3). To study Q.D.s. inR n , one has to supply the subject with new techniques. This is the goal of papers by Shapiro, Khavinson and Shapiro, Sakai, and Gustafsson, where the authors approach the subject inR n by different methods. The main purpose of this paper is to generalize some of the ideas (already known inR 2) toR n (n≥3), and we merely work with unbounded Q.D.

Unbounded quadrature domains in ℝ n (n ≥ 3)

Abstract: As a result of the powerful tools of complex analysis a lot of problems have been solved in the theory of Q.D.s (quadrature domain) inR 2. These problems are almost untouched inR n (n≥3). To study Q.D.s. inR n , one has to supply the subject with new techniques. This is the goal of papers by Shapiro, Khavinson and Shapiro, Sakai, and Gustafsson, where the authors approach the subject inR n by different methods. The main purpose of this paper is to generalize some of the ideas (already known inR 2) toR n (n≥3), and we merely work with unbounded Q.D.

Extreme instability of the horocycle flow

Abstract: Let g be aC 3 negatively curved Riemannian metric on a compact connected orientable surfaceS. LetB be the collection of all metrics resulting from sufficiently small conformal changes of the metricg. (1) Then there is a constantA > 0 such that ifB then the $$\bar d$$ distance between the horocycle flowĥ t (Margulis parametrization) of (S, ĝ) and the rescaled horocycle flowh ct of (S, g) is at leastA (∀c > 0). No other dynamical system is known to have such extreme instability. (2) Fix e > 0. Then there is anN > 0 so that if we are given samples {ξ} 0 {η} 0 which arose from the horocycle flows corresponding to two of the metricsĝ, g ∈B, then either the two samples are $$\bar d$$ farther thanA/2 apart or the two surfaces are closer than e. This holds even if these samples are slightly inaccurate.