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 .     

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.     

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.     

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.     

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.      

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.     

Estimates for the -Neumann problem and nonexistence of C 2 Levi-flat hypersurfaces in

Let be a pseudoconvex domain with C² boundary in , n 2. We prove that the -Neumann operator N exists for square-integrable forms on . Furthermore, there exists a number ₀>0 such that the operators and the Bergman projection are regular in the Sobolev space W ( ) for <₀. The -Neumann operator is used to construct -closed extension on for forms on the boundary b. This gives solvability for the tangential Cauchy-Riemann operators on the boundary. Using these results, we show that there exist no non-zero L²-holomorphic (p, 0)-forms on any domain with C² pseudoconcave boundary in with p > 0 and n 2. As a consequence, we prove the nonexistence of C² Levi-flat hypersurfaces in .This paper is a revision of our preprint (May 2003) formerly titled Estimates for the -Neumann problem and nonexistence of Levi-flat hypersurfaces in where the nonexistence of C², Levi-flat hypersurfaces is proved for >0.All three authors are partially supported by NSF grants.An erratum to this article can be found at     

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 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.     

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.     

<Emphasis Type="Italic">BMO</Emphasis> spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality

We identify the dual space of the Hardy-type space related to the time independent Schrödinger operator =–+V, with V a potential satisfying a reverse Hölder inequality, as a BMO-type space . We prove the boundedness in this space of the versions of some classical operators associated to (Hardy-Littlewood, semigroup and Poisson maximal functions, square function, fractional integral operator). We also get a characterization of in terms of Carlesson measures.Mathematics Subject Classification (2000): 35J10, 42B35, 42B30This research was partially supported by the European Commission, within the IHP Network HARP 2002-2006, contract number HPRN-CT-2001-00273-HARP. Second author also supported by Programa Ramón y Cajal and grant BMF2001-0189 MCyT (Spain). The first and the fifth author supported by grants 5P03A05020 and 5P03A02821 from KBN (Poland) and by Foundation of Polish Sciences Subsidy 3/99. The third and the fourth author supported by grant MCyT BMF2002-04013-C02-02.     

On an eigenvalue problem related to the critical exponent

In this paper we study the eigenvalue problemwhere is a smooth bounded domain, and u is a positive solution of the problemsuch thatwhere S is the best Sobolev constant for the embedding of H¹₀() into L²*(), We prove several estimates for the eigenvalues _i, of (I), i=2,..,N+2 and some qualitative properties of the corresponding eigenfunctions.Supported by M.I.U.R., project Variational methods and nonlinear differential equations.     

The Euler characteristic of local systems on the moduli of genus 3 hyperelliptic curves

For a partition ={₁₂₃0} of non-negative integers, we calculate the Euler characteristic of the local system on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some of low degree, we make a guess for the motivic Euler characteristic of using counting curves over finite fields.Mathematics Subject Classification (1991): 14J15, 20B25     

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.     

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.     

Singular Fano compactifications of (I)

In this paper we study the singular Fano compactifications of with small Gorenstein singularities and determine the structure of such a compactification in the case of the Fano index r 2.Mathematics Subject Classification (2000): 14J45 , 32J05     

Modified Ariki-Koike algebras and cyclotomic <Emphasis Type="Italic">q</Emphasis>-Schur algebras

The modified Ariki-Koike algebra is a variation of the original Ariki-Koike algebra over an integral domain R. When R is a rational function field over the independent parameters, But for general R, is not isomorphic to , and has a simpler structure than . In this paper, we construct a cellular basis of which has a similar property as the cellular basis of introduced by Dipper-James-Mathas. By comparing these two cellular bases, we obtain some estimate on the decomposition numbers of in terms of the decomposition numbers of . We also prove the integral form of the Schur-Weyl reciprocity between a certain quantum algebra U_q and on the tensor space      

Topologically transitive extensions of bounded operators

Let X be any Banach space and T a bounded operator on X. An extension of the pair (X,T) consists of a Banach space in which X embeds isometrically through an isometry i and a bounded operator on such that When X is separable, it is additionally required that be separable. We say that is a topologically transitive extension of (X, T) when is topologically transitive on , i.e. for every pair of non-empty open subsets of there exists an integer n such that is non-empty. We show that any such pair (X,T) admits a topologically transitive extension , and that when H is a Hilbert space, (H,T) admits a topologically transitive extension where is also a Hilbert space. We show that these extensions are indeed chaotic.Mathematics Subject Classification (2000): 47 A 16     

On index one covers of two-dimensional purely log terminal singularities in positive characteristic

Let S be a normal surface over an algebraically closed field k and let be a standard boundary. We consider index 1 covers of the purely log terminal pair (S,). We prove that when S is smooth and char k=p3, then is canonical under some conditions. To prove this, we classify the boundary =(1–1/b_i)D_i which makes (S,) a purely log terminal pair, and then reduce equations defining singularity of to the normal forms of RDP. Unfortunately there are some counterexamples in p=2, and we classify them. These results give partial solutions to the index 1 cover conjecture in positive characteristic.Mathematical Subject Classification (2000): 14E20     

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.