A strict Positivstellensatz for enveloping algebras

Let G be a connected and simply connected real Lie group with Lie algebra . Semialgebraic subsets of the unitary dual of G are defined and a strict Positivstellensatz for positive elements of the universal enveloping algebra of is proved. An erratum to this article is available at .     

Minimal Connected τ-Critical Hypergraphs

A hypergraph is τ-critical if τ(−{E})<τ() for every edge E ∈ , where τ() denotes the transversal number of . We show that if is a connected τ-critical hypergraph, then −{E} can be partitioned into τ()−1 stars of size at least two, for every edge E ∈ . An immediate corollary is that a connected τ-critical hypergraph has at least 2τ()−1 edges. This extends, in a very natural way, a classical theorem of Gallai on colour-critical graphs, and is equivalent to a theorem of Füredi on t-stable hypergraphs. We deduce a lower bound on the size of τ-critical hypergraphs of minimum degree at least two.     

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.     

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     

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      

Double cubics and double quartics

We study a double cover branched over a smooth divisor such that R is cut on V by a hypersurface of degree 2(n−deg(V)), where n ≥ 8 and V is a smooth hypersurface of degree 3 or 4. We prove that X is nonrational and birationally superrigid.     

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.     

Mean convex hulls and least area disks spanning extreme curves

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in such that for any asymptotic curve , there is a canonical mean convex hull containing all minimal planes spanning Γ. Applying this to quasi-Fuchsian manifolds, we show that for any quasi-Fuchsian manifold, there exist a canonical mean convex core capturing all essential minimal surfaces. On the other hand, we also show that for a generic C³-smooth curve in the boundary of C³-smooth mean convex domain in ℝ³, there exist a unique least area disk spanning the curve.     

Positive Projective Space of a <Emphasis Type="Italic">C</Emphasis>*-Algebra with a Tracial Conditional Expectation

Let be a C*-algebra, a subalgebra of its center and Φ: → a tracial faithful conditional expectation. We define the positive projective space as the quotient where G⁺ is the space of positive invertible elements of , and if there exists g invertible in such that a′ = |g|²a. When is abelian, this space is a set of representatives for probability densities equivalent to a given one. The aim of this paper is to endow ℙ⁺ with differentiable structure, a linear connection and a Finsler metric. This is done in a way that given any pair of elements in ℙ⁺, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric. The metric space ℙ⁺ with the given geodesic distance is non positively curved.     

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.     

On some class of sets with multi-sheeted envelope of holomorphy

Let M be a smooth CR-manifold of dimension 2n – 2 contained in the unit sphere We prove that if M is the finite union of boundaries of (n–1)-dimensional analytic varieties and M is globally minimal, then the envelope of holomorphy of is multi-sheeted. In some cases we prove that the number of sheets is exactly two. It is also proved (for 4-dimensional CR-manifolds contained in that the property of being globally minimal (even being of finite type at each point) is generic.Mathematics Subject Classification (2000): 32D10, 32V10, 32V25     

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     

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.     

The Maximum Size of 3-Wise Intersecting and 3-Wise Union Families

Let be an n-uniform hypergraph on 2n vertices. Suppose that and holds for all F₁,F₂,F₃ ∈ . We prove that the size of is at most . The second author was supported by MEXT Grant-in-Aid for Scientific Research (B) 16340027     

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.     

On Resolvable Multipartite G-Designs II

In this paper, it has been proved that and are factorable, where ⊗ denotes wreath product of graphs. As a consequence, a resolvable (k,n,k,2λ) multipartite –design exists for even k. These results generalize the results of Ushio on –factorizations of complete tripartite graphs.     

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.     

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.     

Hölder and Young Inequalities for the Trace of Operators

If A₁, . . . , A_m are positive semidefinite n × n matrices, and if p₁, . . . , p_m are positive real numbers such that then where |X| denotes and tr(X) denotes the trace of X. Moreover, equality holds in either of these inequalities if and only if . This result will be shown to hold as well in unital C*-algebras that have a faithful tracial state.