Showing papers in "Arkiv för Matematik in 2012"

Morrey spaces in harmonic analysis

Abstract: Through a geometric capacitary analysis based on space dualities, this paper addresses several fundamental aspects of functional analysis and potential theory for the Morrey spaces in harmonic analysis over the Euclidean spaces.

Quasiconformality, homeomorphisms between metric measure spaces preserving quasiminimizers, and uniform density property

Abstract: We characterize quasiconformal mappings as those homeomorphisms between two metric measure spaces of locally bounded geometry that preserve a class of quasiminimizers. We also consider quasiconformal mappings and densities in metric spaces and give a characterization of quasiconformal mappings in terms of the uniform density property introduced by Gehring and Kelly.

Whitney coverings and the tent spaces T1, q(γ) for the Gaussian measure

Abstract: We introduce a technique for handling Whitney decompositions in Gaussian harmonic analysis and apply it to the study of Gaussian analogues of the classical tent spaces T1,q of Coifman–Meyer–Stein.

Exhausting domains of the symmetrized bidisc

Abstract: We show that the symmetrized bidisc may be exhausted by strongly linearly convex domains. It shows in particular the existence of a strongly linearly convex domain that cannot be exhausted by domains biholomorphic to convex ones.

Phase transitions for modified Erdős–Rényi processes

Abstract: A fundamental and very well studied region of the Erdős–Renyi process is the phase transition at m∼n/2 edges in which a giant component suddenly appears We examine the process beginning with an initial graph We further examine the Bohman–Frieze process in which edges between isolated vertices are more likely While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class In particular, the growth of the giant component in the barely supercritical region is linear in all cases

Percolation in invariant Poisson graphs with i.i.d. degrees

Abstract: Let each point of a homogeneous Poisson process in R-d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components.

Fonction maximale centrée de Hardy–Littlewood sur les espaces hyperboliques

Abstract: On montre que la fonction maximale centree de Hardy–Littlewood, M, sur les espaces hyperboliques reels $\mathbb{H}^{n} = \mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}}{\raise .17ex\hbox {$\scriptstyle +$}}{\raise .1ex\hbox {$\scriptscriptstyle +$}}{\scriptscriptstyle +}}} \times \mathbb{R}^{n - 1}$ , satisfait l’inegalite de type faible $\| M f \|_{L^{1, \infty}} \leq A (n \log {n}) \| f\|_{1}$ pour toute f∈L 1(ℍ n ), ou A>0 est une constante independante de la dimension n.

A generalization of k-Cohen–Macaulay simplicial complexes

Abstract: For a positive integer k and a non-negative integer t, a class of simplicial complexes, to be denoted by k-CMt, is introduced. This class generalizes two notions for simplicial complexes: being k-Cohen–Macaulay and k-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CMt (=1−CMt) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CMt property under the operation of join of two simplicial complexes. We show that a complex is k-CMt if and only if the links of its non-empty faces are k-CMt−1. We prove that for an integer s≤d, the (d−s−1)-skeleton of a (d−1)-dimensional k-CMt complex is (k+s)-CMt. This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.

Convolution operators in A −∞ for convex domains

Abstract: We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ n and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.

Generalized invertibility of operator matrices

Abstract: In this paper we consider various aspects of generalized invertibility of the operator matrix Open image in new window acting on a Banach space X⊕Y.

Residue currents associated with weakly holomorphic functions

Abstract: We construct Coleff–Herrera products and Bochner–Martinelli type residue currents associated with a tuple f of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case. This include the transformation law, the Poincare–Lelong formula and the equivalence of the Coleff–Herrera product and the Bochner–Martinelli type residue current associated with f when f defines a complete intersection.

Non-liftable Calabi–Yau spaces

Abstract: We construct many new non-liftable three-dimensional Calabi–Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi–Yau space over some number field as introduced by one of us jointily with D. van Straten.

Chern classes of tropical vector bundles

Abstract: We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Most of the definitions presented here for tropical vector bundles will be contained in Torchiani, C., Line Bundles on Tropical Varieties, Diploma thesis, Technische Universitat Kaiserslautern, Kaiserslautern, 2010, for the case of line bundles. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.

Persistence of freeness for Lie pseudogroup actions

Abstract: The action of a Lie pseudogroup \(\mathcal{G}\) on a smooth manifold M induces a prolonged pseudogroup action on the jet spaces J n of submanifolds of M. We prove in this paper that both the local and global freeness of the action of \(\mathcal{G}\) on J n persist under prolongation in the jet order n. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the prolonged pseudogroup actions.

Blaschke condition and zero sets in weighted Dirichlet spaces

Abstract: Let D(μ) be the Dirichlet space weighted by the Poisson integral of the positive measure μ. We give a characterization of the measures μ equal to a countable sum of atoms for which the Blaschke condition is a necessary and sufficient condition for a sequence to be a zero set for D(μ).

Polynomial hulls and proper analytic disks

Abstract: We show how to construct the Perron–Bremermann function by using proper analytic disks. We apply this result to the polynomial hull of a compact set K defined on the boundary of the unit ball.

Infima of superharmonic functions

Abstract: Let Ω be a Greenian domain in ℝ d , d≥2, or—more generally—let Ω be a connected $\mathcal{P}$ -Brelot space satisfying axiom D, and let u be a numerical function on Ω, $u ot\equiv\infty$ , which is locally bounded from below A short proof yields the following result: The function u is the infimum of its superharmonic majorants if and only if each set {x:u(x)>t}, t∈ℝ, differs from an analytic set only by a polar set and $\int u\,d\mu_{x}^{V}\le u(x)$ , whenever V is a relatively compact open set in Ω and x∈V

The topological center of the spectrum of some distal algebras

Abstract: The topological center of the spectrum of the Weyl algebra W, i.e. the norm closure of the algebra generated by the set of functions \(\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i\in\mathbb {N}\}\), is characterized in a recent paper by Jabbari and Namioka (Ellis group and the topological center of the flow generated by the map \(n\mapsto \lambda^{n^{k}}\), to appear in Milan J. Math.). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra Wk, the norm closure of the algebra generated by the set of functions \(\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i=0,1,2,\ldots,k\}\), will be characterized, for all k∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.

Erratum to: The hitting distributions of a half real line for two-dimensional random walks

Abstract: In Theorem 1.1 of my paper [1] (p. 373) there is an erroneous statement. The formula (3) of the theorem expresses the asymptotic form of H x (s), the hitting distribution of the non-positive half line for a random walk on Z started at x. Its first statement, which is true, asserts that (3) holds for x>0. The error is included in the second one, in which the validity of (3) is asserted also for x<0 under the additional moment condition E [|S| log |S|]<∞. For x<0 however, the righthand side of (3) must be multiplied by the ratio (|x|+|s|)/2|x−s|, namely the true statement must read: if E [|S| log |S|]<∞ in addition, then for x<0,

On the scaling limit of loop-erased random walk excursion

Abstract: We use the known convergence of loop-erased random walk to radial SLE(2) to give a new proof that the scaling limit of loop-erased random walk excursion in the upper half-plane is chordal SLE(2). Our proof relies on a version of Wilson’s algorithm for weighted graphs which is used together with a Beurling-type estimate for random walk excursion. We also establish and use the convergence of the radial SLE path to the chordal SLE path as the bulk point tends to a boundary point. In the final section we sketch how to extend our results to more general simply connected domains.

A normality criterion involving rotations and dilations in the argument

Abstract: We show that a family \(\mathcal{F}\) of analytic functions in the unit disk \({\mathbb{D}}\) all of whose zeros have multiplicity at least k and which satisfy a condition of the form $$f^n(z)f^{(k)}(xz) e1$$ for all \(z\in{\mathbb{D}}\) and \(f\in\mathcal{F}\) (where n≥3, k≥1 and 0<|x|≤1) is normal at the origin. The proof relies on a modification of Nevanlinna theory in combination with the Zalcman–Pang rescaling method. Furthermore we prove the corresponding Picard-type theorem for entire functions and some generalizations.

On the holomorphic extension of CR functions from non-generic CR submanifolds of ℂn

Abstract: We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold N of ℂn to complex transversal wedges with edges containing N. We show that given any v∈ℂn∖(TpN+iTpN), there exists a wedge of direction v whose edge contains a neighborhood of p in N, such that any continuous CR function defined locally near p extends holomorphically to that wedge.

Nonexistence of Levi-degenerate hypersurfaces of constant signature in CPn

Abstract: Let M be a smooth hypersurface of constant signature in CP n , n≥3. We prove the regularity for Open image in new window on M in bidegree (0,1). As a consequence, we show that there exists no smooth hypersurface in CP n , n≥3, whose Levi form has at least two zero-eigenvalues.