Showing papers in "Bulletin Des Sciences Mathematiques in 2018"

Toeplitz and asymptotic Toeplitz operators on H2(Dn)

Abstract: We initiate a study of Toeplitz operators and asymptotic Toeplitz operators on the Hardy space H 2 ( D n ) (over the unit polydisc D n in C n ). Our main results on Toeplitz and asymptotic Toeplitz operators can be stated as follows: Let T z i denote the multiplication operator on H 2 ( D n ) by the i-th coordinate function z i , i = 1 , … , n , and let T be a bounded linear operator on H 2 ( D n ) . Then the following hold: (i) T is a Toeplitz operator (that is, T = P H 2 ( D n ) M φ | H 2 ( D n ) , where M φ is the Laurent operator on L 2 ( T n ) for some φ ∈ L ∞ ( T n ) ) if and only if T z i ⁎ T T z i = T for all i = 1 , … , n . (ii) T is an asymptotic Toeplitz operator if and only if T = Toeplitz + compact . The case n = 1 is the well known results of Brown and Halmos, and Feintuch, respectively. We also present related results in the setting of vector-valued Hardy spaces over the unit disc.

On the ideal case of a conjecture of Auslander and Reiten

Abstract: A celebrated conjecture of Auslander and Reiten claims that a finitely generated module M that has no extensions with M ⊕ Λ over an Artin algebra Λ must be projective. This conjecture is widely open in general, even for modules over commutative Noetherian local rings. Over such rings, we prove that a large class of ideals satisfy the extension condition proposed in the aforementioned conjecture of Auslander and Reiten. Along the way we obtain a new characterization of regularity in terms of the injective dimensions of certain ideals.

Schrödinger–Hardy systems involving two Laplacian operators in the Heisenberg group

Abstract: In this paper we first prove existence of nontrivial nonnegative solutions of a Schrodinger–Hardy system in the Heisenberg group, driven by two possibly different Laplacian operators. The main originality of the paper is to work in the Heisenberg group. In fact several new theorems have to be proved in order to overcome the difficulties arising in the new framework, also due to the presence of the Hardy terms and the fact that the nonlinearities do not necessarily satisfy the Ambrosetti–Rabinowitz condition. Finally, we discuss and prove existence even for systems in the Heisenberg group, including critical nonlinear terms.

Structural rigidity of generalised Volterra operators on H-P

Abstract: We show that the non-compact generalised analytic Volterra operators T g , where g ∈ BMOA , have the following structural rigidity property on the Hardy spaces H p for 1 ≤ p ∞ and p ≠ 2 : if T g is bounded below on an infinite-dimensional subspace M ⊂ H p , then M contains a subspace linearly isomorphic to l p . This implies in particular that any Volterra operator T g : H p → H p is l 2 -singular for p ≠ 2 .

Mutation of friezes

Abstract: We study mutations of Conway-Coxeter friezes which are compatible with mutations of cluster-tilting objects in the associated cluster category of Dynkin type A. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. We observe how the frieze can be divided into four distinct regions, relative to the entry at which we want to mutate, where any two entries in the same region obey the same mutation rule. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type A in the sense of Caldero and Chapoton.

Weak differentiability of Wiener functionals and occupation times

Abstract: In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Leao, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t. Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian paths under finite p-variation ( p ≥ 2 ) regularity is also discussed. Under stronger regularity conditions in the sense of finite ( p , q ) -variation, the connection between weak differentiability and two-parameter local time integrals in the sense of Young is established.

Hardy–Littlewood and Pitt's inequalities for Hausdorff operators

Abstract: In this paper we study transformed trigonometric series with Hausdorff averages of Fourier coefficients. We prove Hardy–Littlewood and Pitt's inequalities for such series. The corresponding results for the Hausdorff averages of the Fourier transforms are also obtained.

Twisted argyle quivers and Higgs bundles

Abstract: Ordinarily, quiver varieties are constructed as moduli spaces of quiver representations in the category of vector spaces. It is also natural to consider quiver representations in a richer category, namely that of vector bundles on some complex variety equipped with a fixed sheaf that twists the morphisms. Representations of A-type quivers in this twisted category — known in the literature as “holomorphic chains” — have practical use in questions concerning the topology of the moduli space of Higgs bundles. In that problem, the variety is a Riemann surface of genus at least 2, and the twist is its canonical line bundle. We extend the treatment of twisted A-type quiver representations to any genus using the Hitchin stability condition induced by Higgs bundles and computing their deformation theory. We then focus in particular on so-called “argyle quivers”, where the rank labelling alternates between 1 and integers r i ≥ 1 . We give explicit geometric identifications of moduli spaces of twisted representations of argyle quivers on P 1 using invariant theory for a non-reductive action via Euclidean reduction on polynomials. This leads to a stratification of the moduli space by change of bundle type, which we identify with “collision manifolds” of invariant zeroes of polynomials. We also relate the present work to Bradlow–Daskalopoulos stability and Thaddeus' pullback maps to stable tuples. We apply our results to computing Q -Betti numbers of low-rank twisted Higgs bundle moduli spaces on P 1 , where the Higgs fields take values in an arbitrary ample line bundle. Our results agree with conjectural Poincare series arising from the ADHM recursion formula.

Conjugation orbits of loxodromic pairs in SU(n,1)

Abstract: Let H C n be the n-dimensional complex hyperbolic space and SU ( n , 1 ) be the (holomorphic) isometry group. An element g in SU ( n , 1 ) is called loxodromic or hyperbolic if it has exactly two fixed points on the boundary ∂ H C n . We classify SU ( n , 1 ) conjugation orbits of pairs of loxodromic elements in SU ( n , 1 ) .

Non-emptiness of Brill-Noether Loci over very general quintic hypersurface

Abstract: In this article we study Brill–Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill–Noether loci over very general quintic hypersurface in P 3 , and use the Petri map to produce components of expected dimension.

Invariant measures for the non-periodic two-dimensional Euler equation

Abstract: We construct Gaussian invariant measures for the 2D Euler equation on the plane. We obtain them as the weak limit of those previously considered in [1] for the torus. We show the existence of solution with initial conditions on the support of the measures. Uniqueness and continuity of the velocity flow are proved.

On general multilinear square function with non-smooth kernels

Abstract: In this paper, we obtain some boundedness of multilinear square functions $T$ with non-smooth kernels which extend some known results significantly. We also introduce the multilinear maximal square function $T^*$ and weighted strong and weak type estimates for $T^*$ are given. % As applications, we deduce some weighted estimates for Calder\'on commutator and multilinear Littlewood-paley operator.

Asymptotic behavior of Poisson integrals in a cylinder and its application to the representation of harmonic functions

Abstract: Our first aim in this paper is to deal with asymptotic behavior of Poisson integrals in a cylinder. Next Carleman's formula for harmonic functions in it is also proved. As an application of them, we finally give the integral representation of harmonic functions in a cylinder.

Time fractional linear problems on L2(Rd)

Abstract: In this paper, we propose a theory for linear time fractional PDEs on L 2 ( R d ) , with two time parameters. The order of the time derivatives under consideration is less than 1. We study well-posedness, regularizing effects and dissipative properties. Regarding regularizing effects, we describe quite precisely the equations that have this effect or not. We highlight that, in purely fractional settings, the regularizing effect acts always only up to finite order; unlike to the standard case. Also, we investigate the properties of the three time variables solution operator generated by these PDEs.

An operator-valued kernel associated with a commuting tuple of Hilbert space operators

Abstract: We associate a one parameter family of positive definite E-valued kernels K a , T with any commuting d-tuple T of bounded linear operators on a Hilbert space H, where a is a multi-sequence of non-zero complex numbers and E is an auxiliary Hilbert space. If H a , T denotes the reproducing kernel Hilbert space associated with K a , T , then there exists an isometry U a , T from H a , T into H. It turns out that U a , T is surjective if and only E is a cyclic subspace for T. We apply the above scheme to the commuting toral Cauchy dual d-tuple S t and the constant multi-sequence a t with value 1 (resp. commuting spherical Cauchy dual d-tuple S s and the multi-sequence a s , α : = ( d + | α | − 1 ) ! ( d − 1 ) ! α ! , α ∈ N d ) with E being the joint kernel of S ⁎ to ensure an analytic model for S under some natural assumptions. In particular, the strictly higher dimensional obstruction to the intertwining of U a , S t with S t (resp. the intertwining of U a , S s with S s ) and the multiplication tuple M z is characterized in terms of a kernel condition. These results can be considered as toral and spherical analogs of Shimorin's Theorem (the case of d = 1 ) stating that any left-invertible analytic operator admits an analytic model.

Operator-valued Jacobi parameters and examples of operator-valued distributions

Abstract: In the setting of distributions taking values in a C ⁎ -algebra B , we define generalized Jacobi parameters and study distributions they generate. These include numerous known examples and one new family, of B -valued free binomial distributions, for which we are able to compute free convolution powers. Moreover, we develop a convenient combinatorial method for calculating the joint distributions of B -free random variables with Jacobi parameters, utilizing two-color non-crossing partitions. This leads to several new explicit examples of free convolution computations in the operator-valued setting. Additionally, we obtain a counting algorithm for the number of two-color non-crossing pairings of relative finite depth, using only free probabilistic techniques. Finally, we show that the class of distributions with Jacobi parameters is not closed under free convolution.

Existence of global solutions for a class of vector fields on the three-dimensional torus

Abstract: This work deals with global solvability of a class of vector fields of the form L = ∂ / ∂ t + ( a ( x ) + i b ( x ) ) ( ∂ / ∂ x + λ ∂ / ∂ y ) , where a , b ∈ C ∞ ( T 1 , R ) and λ ∈ R , defined on the three-dimensional torus T 3 ( x , y , t ) ≃ R 3 / 2 π Z 3 . In addition to the interplay between the order of vanishing of the functions a and b, the change of sign of b between two consecutive zeros of a + i b has influence in the global solvability. Also, a Diophantine condition appears in a natural way in our results.

A fundamental differential system of 3-dimensional Riemannian geometry

Abstract: We briefly recall a fundamental exterior differential system of Riemannian geometry and apply it to the case of three dimensions. Here we find new global tensors and intrinsic invariants of oriented Riemannian 3-manifolds. In particular, we develop the study of ∇Ric. The exterior differential system leads to a remarkable Weingarten type equation for immersed surfaces in hyperbolic 3-space. A new independent proof for low dimensions of the structural equations gives new insight on the intrinsic exterior differential system.

Stability of sheaves over Quot schemes

Abstract: Let C be a smooth projective curve over an algebraically closed field k. Let Q ( r , d ) be the Quot scheme of degree d zero dimensional quotients of O C r . Let A ( r , d ) be the kernel of the universal quotient map over C × Q ( r , d ) and let π Q be the second projection from C × Q ( r , d ) to Q ( r , d ) . Then, we prove that with respect to certain natural polarisations on C × Q ( r , d ) and Q ( r , d ) , A ( r , d ) and ( π Q ) ⁎ A ( r , d ) are stable. A similar statement over Flag scheme of torsion quotients of O r C is also proved.

k-Uniform rotundity is equivalent to k-uniform convexity

Abstract: R. Geremia and F. Sullivan proved that 2-uniform rotundity is equivalent to 2-uniform convexity. Pei-Kee Lin extended this result for any integer k ≥ 1 . This paper gives a different proof for the equivalence of k-uniform rotundity and k-uniform convexity for any integer k ≥ 1 .

A propos de l'algèbre de Hopf des mots tassés WMat

Abstract: Resume Dans cet article, on etudie l'algebre de Hopf des mots tasses WMat introduite par Duchamp, Hoang-Nghia et Tanasa. Pour cela on commence par considerer WMat a proprement parler (absence de coliberte et description du dual). Puis, on s'interesse a une sous-algebre de Hopf de permutations, notee S H , dont le dual S H ⊛ est muni d'une structure de quadri-algebre et donc d'une double structure d'algebre dendriforme. On introduit par la suite ISPW, une algebre de Hopf de mots tasses stricts croissants. Son caractere cocommutatif pousse a s'interesser a ses elements primitifs. On en decrit quelques familles. On montre ensuite que ISPW et l'algebre de Hopf NSym des fonctions symetriques non commutatives sont isomorphes. On definit par la suite une algebre de Hopf quotient de compositions etendues C e . Celle-ci n'est pas cocommutative mais ses elements primitifs sont lies a ceux de ISPW. De plus, on exprime C e comme un coproduit semi-direct d'algebres de Hopf. Cette construction met a jour une coaction permettant par la suite de definir deux actions de groupes. On termine par la construction d'un isomorphisme explicite entre ISPW ⊛ et QSym conduisant a un isomorphisme explicite entre ISPW et NSym.

Born–Jordan pseudodifferential operators and the Dirac correspondence: Beyond the Groenewold–van Hove theorem

Abstract: Quantization procedures play an essential role in operator theory, time–frequency analysis and, of course, in quantum mechanics. Roughly speaking the basic idea, due to Dirac, is to associate to any symbol, or observable, a ( x , ξ ) an operator Op ( a ) , according to some axioms dictated by physical considerations. This led to the introduction of a variety of quantizations. They all agree when the symbol a ( x , ξ ) = f ( x ) depends only on x or a ( x , ξ ) = g ( ξ ) depends only on ξ: Op ( f ⊗ 1 ) u = f u , Op ( 1 ⊗ g ) u = F − 1 ( g F u ) where F stands for the Fourier transform. Now, Dirac aimed at finding a quantization satisfying, in addition, the key correspondence [ Op ( a ) , Op ( b ) ] = i Op ( { a , b } ) where [ , ] stands for the commutator and { , } for the Poisson brackets, which would represent a tight link between classical and quantum mechanics. Unfortunately, the famous Groenewold–van Hove theorem states that such a quantization does not exist, and indeed most quantization rules satisfy this property only approximately. In this work we show that the above commutator rule in fact holds for the Born–Jordan quantization, at least for symbols of the type f ( x ) + g ( ξ ) . Moreover we will prove that, remarkably, this property completely characterizes this quantization rule, making it the quantization which best fits the Dirac dream.

Sharp seasonal threshold property for cooperative population dynamics with concave nonlinearities

Abstract: We consider a biological population whose environment varies periodically in time, exhibiting two very different " seasons " : one is favorable and the other one is unfavorable. For monotone differential models with concave nonlinearities, we address the following question: the system's period being fixed, under what conditions does there exist a critical duration for the unfavorable season? By " critical duration " we mean that above some threshold, the population cannot sustain and extincts, while below this threshold, the system converges to a unique periodic and positive solution. We term this a " sharp seasonal threshold property " (SSTP, for short). Building upon a previous result, we obtain sufficient conditions for SSTP in any dimension and apply our criterion to a two-dimensional model featuring juvenile and adult populations of insects.

Relaxation of nonconvex unbounded integrals with general growth conditions in Cheeger–Sobolev spaces

Abstract: We study relaxation of nonconvex integrals of the calculus of variations in the setting of Cheeger–Sobolev spaces when the integrand does not have polynomial growth and can take infinite values.

On the center criterion of planar quasi-homogeneous polynomial differential systems

Abstract: In this paper, we characterize the centers of the quasi-homogeneous planar polynomial differential systems of any degree n ∈ N . First, we prove that the system does not admit centers if n is even, then present the sufficient and necessary conditions that the system admits a center if n is odd. Furthermore, we provide a simple method to obtain such these systems with center for a given order.

Density of the span of powers of a function à la Müntz-Szasz

Abstract: The aim of this paper is to establish density properties in $L^p$ spaces of the span of powers of functions $\{\psi^\lambda\,:\lambda\in\Lambda\}$, $\Lambda\subset\N$ in the spirit of the M\"untz-Sz\'asz Theorem. As density is almost never achieved, we further investigate the density of powers and a modulation of powers $\{\psi^\lambda,\psi^\lambda e^{i\alpha t}\,:\lambda\in\Lambda\}$. Finally, we establish a M\"untz-Sz\'asz Theorem for density of translates of powers of cosines $\{\cos^\lambda(t-\theta_1),\cos^\lambda(t-\theta_2)\,:\lambda\in\Lambda\}$. Under some arithmetic restrictions on $\theta_1-\theta_2$, we show that density is equivalent to a M\"untz-Sz\'asz condition on $\Lambda$ and we conjecture that those arithmetic restrictions are not needed. Some links are also established with the recently introduced concept of Heisenberg Uniqueness Pairs.

Brauer group of the moduli spaces of stable vector bundles of fixed determinant over a smooth curve

Abstract: Let X be an irreducible smooth projective curve, defined over an algebraically closed field k, of genus at least three and L a line bundle on X. Let M X ( r , L ) be the moduli space of stable vector bundles on X of rank r and determinant L with r ≥ 2 . We prove that the Brauer group Br ( M X ( r , L ) ) is cyclic of order g . c . d . ( r , degree ( L ) ) . We also prove that Br ( M X ( r , L ) ) is generated by the class of the projective bundle obtained by restricting the universal projective bundle. These results were proved earlier in [1] under the assumption that k = C .

Products of weighted multiple zeta functions

Abstract: It is well known that the multiple zeta functions satisfy the stuffle relation. Recently it is proved (see J. Mehta et al. (2016) [3] ) that a weighted variant of the multiple zeta function satisfies the shuffle relation. In this article we characterize all the weighted multiple zeta functions which satisfy the stuffle relation (respectively shuffle relation).