Holomorphic dynamics near germs of singular curves

Let M be a two dimensional complex manifold, p ∈ M and a germ of holomorphic foliation of M at p. Let be a germ of an irreducible, possibly singular, curve at p in M which is a separatrix for . We prove that if the Camacho-Sad-Suwa index Ind then there exists another separatrix for at p. A similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points.     

Summing norms of identities between unitary ideals

Based on abstract interpolation, we prove asymptotic formulae for the (F,2)-summing norm of inclusions id: , where E and F are two Banach sequence spaces. Here, stands for the unitary ideal of operators on the n-dimensional Hilbert space whose singular values belong to E, and for the Hilbert-Schmidt operators. Our results are noncommutative analogues of results due to Bennett and Carl, as well as their recent generalizations to Banach sequence spaces. As an application, we give lower and upper estimates for certain s-numbers of the embeddings id: and id: . In the concluding section, we finally consider mixing norms. The second named author was supported by KBN Grant 2 P03A 042 18.     

Hausdorff measures, capacities and compact composition operators

It is shown that there exist analytic self-maps ϕ of the unit disc inducing compact composition operators on the Hardy space , 1 ≤ p < ∞ such that the Hausdorff dimension of the set is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces with 0 < α < 1. As a consequence, we deduce that there exist symbols ϕ inducing compact composition operators on such that the α-capacity of E_ϕ is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on . First author is partially supported by Plan Nacional I+D grant no. BFM2003-00034, and Gobierno de Aragón research group Análisis Matemático y Aplicaciones, ref. DGA E-64 . Second author is partially supported by Plan Nacional I+D grant no. BFM2002-00571 and Junta de Andalucía RNM-314.     

Continuity of eigenvalues of subordinate processes in domains

Let X={X_t,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S⁽ⁿ⁾={S⁽ⁿ⁾_t, t ≥ 0} be a subordinator with Laplace exponent ϕ_n and S={S_t,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S⁽ⁿ⁾. In this paper we consider the subordinate processes and and their subprocesses and X^ϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of and X^ϕ,D are all discrete, with being the eigenvalues of the generator of and being the eigenvalues of the generator of X^ϕ,D. We show that, if lim_n→∞ϕ_n(λ)=ϕ(λ) for every λ>0, then The research of this author is supported in part by NSF Grant DMS-0303310. The research of this author is supported in part by a joint US-Croatia grant INT 0302167.     

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.      

Spectral multipliers for sub-Laplacians with drift on Lie groups

We study spectral multipliers of right invariant sub-Laplacians with drift on a connected Lie group G. The operators we consider are self-adjoint with respect to a positive measure , whose density with respect to the left Haar measure λ_G is a nontrivial positive character of G. We show that if p≠2 and G is amenable, then every spectral multiplier of extends to a bounded holomorphic function on a parabolic region in the complex plane, which depends on p and on the drift. When G is of polynomial growth we show that this necessary condition is nearly sufficient, by proving that bounded holomorphic functions on the appropriate parabolic region which satisfy mild regularity conditions on its boundary are spectral multipliers of . Work partially supported by the EC HARP Network “Harmonic Analysis and Related Problems”, the Progetto Cofinanziato MURST “Analisi Armonica” and the Gruppo Nazionale INdAM per l'Analisi Matematica, la Probabilità e le loro Applicazioni. Part of this work was done while the second and the third author were visiting the “Centro De Giorgi” at the Scuola Normale Superiore di Pisa, during a special trimester in Harmonic Analysis. They would like to express their gratitude to the Centro for the hospitality.     

Partial regularity for stationary harmonic maps at a free boundary

Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H¹(, ) with a free boundary condition of the type u(∂) ⊂ Γ, given a submanifold Γ⊂. We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold Γ.     

Proper holomorphic embeddings of finitely and some infinitely connected subsets of into

We show that any finitely connected domain can be properly embedded into . For some sequences can also be properly embedded into .     

Area density and regularity for soap film-like surfaces spanning graphs

For a given boundary Γ in Rⁿ consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface Σ spanning Γ. Σ is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature (Γ) for such graphs, or nets, Γ. We show that 2π times the area density of Σ at any point is less than or equal to (Γ). For n=3, these density estimates imply, for example, that if (Γ)≤3.649π, then the only possible singularities of a piecewise smooth (M,0,δ)-minimizing set Σ are curves, along which three smooth sheets of Σ meet with equal angles of 120^°. Second author supported in part by NSF grant 00-71862.     

Regularity theorems and energy identities for Dirac-harmonic maps

We study Dirac-harmonic maps from a Riemann surface to a sphere We show that a weakly Dirac-harmonic map is in fact smooth, and prove that the energy identity holds during the blow-up process.The research of QC and JYL was partially supported by NSFC. QC was also partially supported by the FOK Yingtung Education Foundation.     

A schlichtness theorem for envelopes of holomorphy

Let Ω be a domain in . We prove the following theorem. If the envelope of holomorphy of Ω is schlicht over Ω, then the envelope is in fact schlicht. We provide examples showing that the conclusion of the theorem does not hold in , n>2. Additionally, we show that the theorem cannot be generalized to provide information about domains in whose envelopes are multiply sheeted.     

Ribbon tableaux and the Heisenberg algebra

In [LLT] Lascoux, Leclerc and Thibon introduced symmetric functions which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these functions in analogy with Schur functions. In particular we will describe: a Pieri and dual-Pieri formula for ribbon functions, a ribbon Murnaghan-Nakayama formula, ribbon Cauchy and dual Cauchy identities, and a -algebra isomorphism _n:(q)(q) which sends each to .Our study of the functions will be connected to the Fock space representation F of via a linear map :F(q) which sends the standard basis of F to the ribbon functions. Kashiwara, Miwa and Stern [KMS] have shown that a copy of the Heisenberg algebra H acts on F commuting with the action of . Identifying the Fock Space of H with the ring of symmetric functions (q) we will show that is in fact a map of H-modules with remarkable properties. The study of this map will lead to our identities concerning ribbon tableaux generating functions. We will also give a combinatorial proof that the ribbon Murnaghan-Nakayama and Pieri rules are formally equivalent.     

Mean-value theorems for multiplicative functions on Beurling’s generalized integers (I)

Halász’s general mean-value theorem for multiplicative functions on ℕ is classical in probabilistic number theory. We extend this theorem to functions f, defined on a set of generalized integers associated with a set of generalized primes in Beurling’s sense, which satisfies Halász’s conditions, in particular,Assume that the distribution function N(x) of satisfieswith γ>γ₀, where ρ₁<ρ₂<···<ρ_m are constants with ρ_m≥1 and A₁,···,A_m are real constants with A_m>0. Also, assume that the Chebyshev function ψ(x) of satisfieswith M>M₀. Then the asymptoticimplieswhere τ is a positive constant with τ≥1 and L(u) is a slowly oscillating function with |L(u)|=1.     

On the category of modules of Gorenstein dimension zero

Let R be a commutative noetherian henselian non-Gorenstein local ring of depth zero. Denote by modR the category of all finitely generated R-modules, and by the full subcategory of modR consisting of all R-modules of Gorenstein dimension zero. We prove in this paper that if contains a non-free module, then it is not precovering in modR, in particular, there exist infinitely many isomorphism classes of indecomposable R-modules of Gorenstein dimension zero.     

Mukai flops and deformations of symplectic resolutions

We prove that two projective symplectic resolutions of are connected by Mukai flops in codimension 2 for a finite sub-group G <Sp(2n). It is also shown that two projective symplectic resolutions of are deformation equivalent.     

All double coverings of cyclotomic fields

A double covering of a Galois extension K/F in the sense of [3] is an extension /K of degree ≤2 such that /F is Galois. In this paper we determine explicitly all double coverings of any cyclotomic extension over the rational number field in the complex number field. We get the results mainly by Galois theory and by using and modifying the results and the methods in [2] and [3]. Project 10571097 supported by NSFC     

Canonical maps to twisted rings

If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring , which is surjective in large degree. This result is a key step in the study of projectively simple rings. The proof relies on some results concerning the growth of graded rings which are of independent interest. D. Rogalski was partially supported by NSF grant DMS-0202479. J. J. Zhang was partially supported by NSF grant DMS-0245420 and Leverhulme Research Interchange Grant F/00158/X (UK).     

On certain categories of modules for affine Lie algebras

In this paper, we exploit basic formal variable techniques to study certain categories of modules for an (untwisted) affine Lie algebra , motivated by Chari-Pressleys work on certain integrable modules. We define and study two categories and of -modules using generating functions, where is proved to contain the well known evaluation modules and to unify highest weight modules, evaluation modules and their tensor product modules. We classify integrable irreducible -modules in categories and and we determine the isomorphism classes of those irreducible modules. Finally we prove a result that relates fusion rules in the context of vertex operator algebras with integrable irreducible modules of Chari-Pressley.in final form: 12 November 2003Partially supported by a NSA grant and a grant from Rutgers Research Council.     

Some remarks on the Hardy-Littlewood maximal function on variable <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We show that any pointwise multiplier for BMO(ℝⁿ) generates a function p from the class (ℝⁿ) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L^p space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to (ℝⁿ).     

The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations

For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification points to the zeta-function. In fact, if is the ring of Eisenstein integers, then the Selberg zeta-function of contains ramification points and is the sixth-root of a meromorphic function.