Showing papers in "Publicacions Matematiques in 2012"

Sharp norm inequalities for commutators of classical operators

Abstract: We prove several sharp weighted norm inequalities for commutators of classical operators in harmonic analysis. We found suffcient Ap-bump conditions on pairs of weights (u; v) such that [b; T], b 2 BMO and T a singular integral operator (such as the Hilbert or Riesz transforms), maps Lp(v) into Lp(u). Because of the added degree of singularity, the commutators require a \double log bump" as opposed to that of singular integrals, which only require single log bumps. For the fractional integral operator I we nd the sharp one-weight bound on [b; I ], b 2 BMO, in terms of the Ap;q constant of the weight. We also prove sharp two-weight bounds for [b; I ] analogous to those of singular integrals. We prove two-weight weak type inequalities for [b; T] and [b; I ] for pairs of factored weights. Finally we construct several examples showing our bounds are sharp.

Elliptic obstacle problems with measure data : potentials and low order regularity

Abstract: We consider obstacle problems with measure data related to elliptic equations of p-Laplace type, and investigate the connections between low order regularity properties of the solutions and non-linear potentials of the data. In particular, we give pointwise estimates for the solutions in terms of Wol potentials and address the questions of boundedness and continuity of the solution.

An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange

Abstract: The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those au- tomorphisms of G that can be extended, in a functorial manner, to all groups H given with homomorphisms G ! H: (Precise statement in x1.) The group of such extended systems of automorphisms, unlike the group of inner automorphisms of G itself, is always isomorphic to G: A similar characterization holds for inner automorphisms of an associative algebra R over a eld K; here the group of functorial systems of automorphisms is isomorphic to the group of units of R modulo the units of K: If one looks at the above functorial extendibility property for endomorphisms, rather than just automorphisms, then in the group case, the only additional example is the trivial endomorphism; but in the K-algebra case, a construction unfamiliar to ring theorists, but known to functional analysts, also arises. Systems of endomorphisms with the same functoriality property are examined in some other categories; other uses of the phrase \inner endomorphism" in the literature, some overlapping the one introduced here, are noted; the concept of an inner derivation of an associative or Lie algebra is looked at from the same point of view, and the dual concept of a \co-inner" endomorphism is briey examined. Several open questions are noted.

On $D(-1)$-Quadruples

Abstract: Quadruples (a; b; c; d) of positive integers a < b < c < d with the property that the product of any two of them is one more than a perfect square are studied. Improved lower and upper bounds for the entries b and c are established. As an application of these results, a bound for the number of such quadruples is obtained.

Un théorème de point fixe sur les espaces Lp

Abstract: We establish a fixed point theorem for group actions on Lp-spaces, which generalizes a theorem of Zuk and of Ballmann-Swiatkowski to the case p ≠ 2.

Boundedness of rough integral operators on Triebel-Lizorkin spaces

Abstract: We prove the boundedness of several classes of rough integral operators on Triebel-Lizorkin spaces. Our results represent improvements as well as natural extensions of many previously known results.

Thompson’s group is the orientation-preserving automorphism group of a cellular complex

Abstract: We consider a planar surface ∑ of in nite type which has Thompson's group T as asymptotic mapping class group. We construct the asymptotic pants complex C of ∑ and prove that the group T acts transitively by automorphisms on it. Finally, we establish that the automorphism group of the complex C is an extension of the Thompson group T by Z=2Z.

Intermediaries in bredon (co)homology and classifying spaces

Abstract: For certain contractible G-CW-complexes and F a family of subgroups of G, we construct a spectral sequence converging to the F-Bredon cohomology of G with E1-terms given by the F-Bredon cohomology of the stabilizer subgroups. As applications, we obtain several corollaries concerning the cohomological and geometric dimensions of the classifying space EFG. We also introduce, for any subgroup closed class of groups F, a hierarchically de ned class of groups and show that if a group G is in this class, then G has finite F ∩ G-Bredon (co)homological dimension if and only if G has jump F ∩ G-Bredon (co)homology.

A geometric and stochastic proof of the twist point theorem

Abstract: In this paper we will give a proof of the McMillan twist point theorem using geometry, potential theory and Ito's formula but not the Riemann mapping theorem.

Isolated singularities of binary differential equations of degree $n$

Abstract: We study isolated singularities of binary differential equations of degree n which are totally real. This means that at any regular point, the associated algebraic equation of degree n has exactly n different real roots (this generalizes the so called positive quadratic differential forms when n = 2). We introduce the concept of index for isolated singularities and generalize Poincar´e-Hopf theorem and Bendixson formula. Moreover, we give a classification of phase portraits of the n-web around a generic singular point. We show that there are only three types, which generalize the Darbouxian umbilics D1, D2 and D3.

A degree problem for two algebraic numbers and their sum

Abstract: For all but one positive integer triplet (a; b; c) with a < b < c and b < 6, we decide whether there are algebraic numbers α,β and γ of degrees a, b and y, respectively, such that α+β+γ = 0. The undecided case (6; 6; 8) will be included in another paper. These results imply, for example, that the sum of two algebraic numbers of degree 6 can be of degree 15 but cannot be of degree 10. We also show that if a positive integer triplet (a; b; c) satisfies a certain triangle-like inequality with respect to every prime number then there exist algebraic numbers α,β γ of degrees a, b, c such that α+β+γ = 0. We also solve a similar problem for all (a; b; c) with a < b < c and b <6 by finding for which a, b, c there exist number fields of degrees a and b such that their compositum has degree c. Further, we have some results on the multiplicative version of the first problem, asking for which triplets (a; b; c) there are algebraic numbers and α, β and γ of degrees a, b and c, respectively, such that αβγ = 1.

Boundary values in range spaces of co-analytic truncated Toeplitz operators

Abstract: Functions in backward shift invariant subspaces have nice analytic continuation properties outside the spectrum of the inner function de ning the space. Inside the spectrum of the inner function, Ahern and Clark showed that under some distribution condition on the zeros and the singular measure of the inner function, it is possible to obtain non-tangential boundary values of every function in the backward shift invariant subspace as well as for their derivatives up to a certain order. Here we will investigate, at least when the inner function is a Blaschke product, the non- tangential boundary values of the functions of the backward shift invariant subspace after having applied a co-analytic (truncated) Toeplitz operator. There appears to be a smoothing effect.

Green currents for quasi-algebraically stable meromorphic self-maps of Pk

Abstract: We construct a canonical Green current Tf for every quasi-algebraically stable meromorphic self-map f of Pk such that its rst dynamical degree 1(f) is a simple root of its characteristic polynomial and that λ1(f) > 1: We establish a functional equation for Tf and show that the support of Tf is contained in the Julia set, which is thus non empty.

Some non-amenable groups

Abstract: We generalise a result of R. Thomas to establish the non-vanishing of the first L2 Betti number for a class of nitely generated groups.

On non-commuting sets in finite soluble CC-groups

Abstract: Lower bounds for the number of elements of the largest non-commuting set of a finite soluble group with a CC subgroup are considered in this paper.

Asymptotics for the minimum Riesz energy and best-packing on sets of finite packing premeasure

Abstract: We show that for every compact set A ⊂ Rm of finite α -dimensional packing premeasure 0 α ) coincides with the outer measure of A constructed from this limit by method I. The asymptotic behavior of the discrete minimum energy on compact subsets of a self-similar set K satisfying the open set condition is also studied for s greater than the Hausdor dimension of K. In addition, similar problems are studied for the best-packing radius.

On the power pseudovariety

Abstract: The pseudovariety PCS which is generated by all power semigroups of finite completely simple semigroups is characterized in various ways. For example, the equalities PCS = Jm CS = BGm RB are established. This resolves a problem raised by Kad'ourek and leads to several transparent algorithms for deciding membership in PCS.

On the Power Pseudovariety $\mathbf{PCS}$

Abstract: Some new semantic and syntactic characterizations of the members of the power pseudovariety $\mathbf{PCS}$ are obtained. This leads in particular to new algorithms for deciding membership in $\mathbf{PCS}$.

A mi-chemin entre analyse complexe et superanalyse

Abstract: In the framework of superanalysis we get a functions theory close to complex analysis, under a suitable condition (A) on the real superalgebras in consi- deration (this condition is a generalization of the classical relation 1 + i2 = 0 in C). Under the condition (A), we get an integral representation formula for the super- di erentiable functions. We deduce properties of the superdi erentiable functions: analyticity, a result of separated superdi erentiability, a Liouville theorem and a continuation theorem of Hartogs-Bochner type.