The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

A new irreducible component of the moduli space of stable Godeaux surfaces

We construct from a general del Pezzo surface of degree 1 a Gorenstein stable surface X with \({K_X^2=1}\) and p _g (X) = q(X) = 0. These surfaces are not smoothable but give an open subset of an irreducible component of the moduli space of stable Godeaux surfaces. In a particular example we also compute the canonical ring explicitly and discuss the behaviour of pluricanonical maps.     

Toeplitz Operators from One Fock Space to Another

In this paper, we study Toeplitz operators T _μ from one Fock space \({F^{p}_{\alpha}}\) to another \({F^{q}_{\alpha}}\) for 1 < p, q < ∞ with positive Borel measures μ as symbols. We characterize the boundedness (and compactness) of \({T_\mu: F^{p}_{\alpha} \to F^{q}_{\alpha}}\) in terms of the averaging function \({\widehat{\mu}_r}\) and the t-Berezin transform \({\widetilde{\mu}_t}\) respectively. Quite differently from the Bergman space case, we show that T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for some p ≤ q if and only if T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for all p ≤ q. In order to prove our main results on T _μ , we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on C ⁿ .     

Absolute Continuity and Singularity of Two Probability Measures on a Filtered Space

Let μ and ν be fixed probability measures on a filtered space \((\varOmega, \mathcal{F}, \allowbreak(\mathcal{F}_{t} )_{t\in \mathbf{R}^{+} } )\). Denote by μ _T and ν _T (respectively, μ _T? and ν _T?) the restrictions of the measures μ and ν on \(\mathcal{F}_{T} \) (respectively, on \(\mathcal{F}_{T-} \)) for a stopping time T. We find the Hahn decomposition of μ _T and ν _T using the Hahn decomposition of the measures μ, ν and the Hellinger process h _t in the strict sense of order \(\frac{1}{2}\). The norm of the absolutely continuous component of μ _T? with respect to ν _T? is computed in terms of density processes and Hellinger integrals.     

Triangulations and Volume Form on Moduli Spaces of Flat Surfaces

In this paper, we study the moduli spaces of flat surfaces with cone singularities verifying the following property: there exists a union of disjoint geodesic tree on the surface such that the complement is a translation surface. Those spaces can be viewed as deformations of the moduli spaces of translation surfaces in the space of flat surfaces. We prove that such spaces are quotients of flat complex affine manifolds by a group acting properly discontinuously, and preserving a parallel volume form. Translation surfaces can be considered as a special case of flat surfaces with erasing forest, in this case, it turns out that our volume form coincides with the usual volume form (which are defined via the period mapping) up to a multiplicative constant. We also prove similar results for the moduli space of flat metric structures on the n-punctured sphere with prescribed cone angles up to homothety. When all the angles are smaller than 2π, it is known (cf. [T]) that this moduli space is a complex hyperbolic orbifold. In this particular case, we prove that our volume form induces a volume form which is equal to the complex hyperbolic volume form up to a multiplicative constant.     

Asymptotic equalities for best approximations for classes of infinitely differentiable functions defined by the modulus of continuity

We obtain asymptotic estimates for best approximations by trigonometric polynomials in the metric of the space C(L_p) for classes of periodic functions expressible as convolutions of kernels Ψ_β with Fourier coefficients decreasing to zero faster than any power sequence, and with functions ? ∈ C (? ∈ L_p) whose moduli of continuity do not exceed the given majorant of ω(t). It is proved that, in the spaces C and L₁, for convex moduli of continuity ω(t), the obtained estimates are asymptotically sharp.     

Oscillations of cusp form coefficients in exponential sums

Let f(z)=∑ _n=1 ^∞ λ(n)n ^(κ?1)/2 e(nz) be a holomorphic cusp form of weight κ for the full modular group SL ₂(?) and let μ(n) be the Möbius function. In this paper, we are concerned with the sum

$S(\alpha,X)=\sum _{n\leq X}\mu (n)\lambda(n)e(\alpha \sqrt{n}),\quad 0\neq \alpha \in \mathbb{R}.$

It is proved that, unconditionally, \(S(\alpha,X)\ll X^{\frac{5}{6}}(\log X)^{20}\), where the implied constant depends only on α and the cusp form f.

     

A Tannakian Interpretation of the Elliptic Infinitesimal Braid Lie Algebras

Let n ≥?1. The pro-unipotent completion of the pure braid group of n points on a genus 1 surface has been shown to be isomorphic to an explicit pro-unipotent group with graded Lie algebra using two types of tools: (a) minimal models (Bezrukavnikov), (b) the choice of a complex structure on the genus 1 surface, making it into an elliptic curve E, and an appropriate flat connection on the configuration space of n points in E (joint work of the authors with D. Calaque). Following a suggestion by P. Deligne, we give an interpretation of this isomorphism in the framework of the Riemann-Hilbert correspondence, using the total space E ^# of an affine line bundle over E, which identifies with the moduli space of line bundles over E equipped with a flat connection.     

Perturbation of chains of de Branges spaces

We investigate the structure of the set of de Branges spaces of entire functions which are contained in a space L²(μ). Thereby, we follow a perturbation approach. The main result is a growth dependent stability theorem. Namely, assume that measures μ₁ and μ₂ are close to each other in a sense quantified relative to a proximate order. Consider the sections of corresponding chains of de Branges spaces C₁ and C₂ which consist of those spaces whose elements have finite (possibly zero) type with respect to the given proximate order. Then either these sections coincide or one is smaller than the other but its complement consists of only a (finite or infinite) sequence of spaces.

Among other situations, we apply—and refine—this general theorem in two important particular situations

1. (1)
   the measures μ₁ and μ₂ differ in essence only on a compact set; then stability of whole chains rather than sections can be shown
    
2. (2)
   the linear space of all polynomials is dense in L²(μ₂); then conditions for density of polynomials in the space L²(μ₂) are obtained.
    

In the proof of the main result, we employ a method used by P. Yuditskii in the context of density of polynomials. Another vital tool is the notion of the index of a chain, which is a generalisation of the index of determinacy of a measure having all power moments. We undertake a systematic study of this index, which is also of interest on its own right.

     

Uniform Homeomorphisms of Unit Spheres and Property H of Lebesgue–Bochner Function Spaces

Assume that the unit spheres of Banach spaces X and Y are uniformly homeomorphic.Then we prove that all unit spheres of the Lebesgue–Bochner function spaces L_p(μ, X) and L_q(μ, Y)are mutually uniformly homeomorphic where 1 ≤ p, q ∞. As its application, we show that if a Banach space X has Property H introduced by Kasparov and Yu, then the space L_p(μ, X), 1 ≤ p ∞,also has Property H.     

Minimal Translation Hypersurfaces in Euclidean Space

The graph of a function f defined in some open set of the Euclidean space of dimension (p + q) is said to be a translation graph if f may be expressed as the sum of two independent functions ? and ψ defined in open sets of the Euclidean spaces of dimension p and q, respectively. We obtain a useful expression for the mean curvature of the graph of f in terms of the Laplacian, the gradient of ? and ψ as well as of the mean curvatures of their graphs. We study translation graphs having zero mean curvature, that is, minimal translation graphs, by imposing natural conditions on ? and ψ, like harmonicity, minimality and eikonality (constant norm of the gradient), giving several examples as well as characterization results.     

On the Number of Vedernikov–Ein Irreducible Components of the Moduli Space of Stable Rank 2 Bundles on the Projective Space

We propose a method for finding the exact number of Vedernikov–Ein irreducible components of the first and second types in the moduli space M(0, n) of stable rank 2 bundles on the projective space P³ with Chern classes c₁ = 0 and c₂ = n ≥ 1. We give formulas for the number of Vedernikov–Ein components and find a criterion for their existence for arbitrary n ≥ 1.     

Equiconvergence of spectral decompositions for the Dirac system with potential in Lebesgue spaces

The problem of equiconvergence of spectral decompositions corresponding to the systems of root functions of two one-dimensional Dirac operators ?_P,U and ?_0,U with potential P summable on a finite interval and Birkhoff-regular boundary conditions is analyzed. It is proved that in the case of P ∈ L_?[0, π], ? ∈ (1,∞], equiconvergence holds for every function f ∈ L_μ[0, π], μ ∈ [1,∞], in the norm of the space L_ν[0, π], ν ∈ [1,∞], if the indices ?, μ, and ν satisfy the inequality 1/? + 1/μ ? 1/ν ≤ 1 (except for the case when ? = ν = ∞ and μ = 1). In particular, in the case of a square summable potential, the uniform equiconvergence on the interval [0, π] is proved for an arbitrary function f ∈ L₂[0, π].     

A new characterization of RBMO(<Emphasis Type="Italic">μ</Emphasis>) by John-Strömberg sharp maximal functions

Let μ be a nonnegative Radon measure on ? ^d which only satisfies μ (B(x, r)) ? C ₀ r ⁿ for all x ∈ ? ^d , r > 0, with some fixed constants C ₀ > 0 and n ∈ (0, d]. In this paper, a new characterization for the space RBMO(μ) of Tolsa in terms of the John-Strömberg sharp maximal function is established.     

On a conjecture of Bulman-Fleming and Gilmour

If S is a monoid, the set S×S equipped with componentwise S-action is called the diagonal act of S and is denoted by D(S). We prove the following theorem: the right S-act S ⁿ (1≠n∈?) is (principally) weakly flat if and only if \(\prod _{i=1}^{n}A_{i}\) is (principally) weakly flat where A _i , 1≤i≤n are (principally) weakly flat right S-acts, if and only if the diagonal act D(S) is (principally) weakly flat. This gives an answer to a conjecture posed by Bulman-Fleming and Gilmour (Semigroup Forum 79:298–314, 2009). Besides, we present a fair characterization of monoids S over which the diagonal act D(S) is (principally) weakly flat and finally, we impose a condition on D(S) in order to make S a left PSF monoid.     

Principal Functions for Bi-free Central Limit Distributions

We find the principal function of the completely non-normal operator l(v₁) + l(v₁)* + i(r(v₂) + r(v₂)*) on a subspace of the full Fock space \({\mathcal{F}}({\mathcal{H}})\) which arises from a bi-free central limit distribution. As an application, we find the essential spectrum of this operator.     

Topological entropy of sets of generic points for actions of amenable groups

Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered F?lner sequence {Fn} in G with limn→+∞|Fn|/log n= ∞, we prove the following result:h_top~B(G_μ, {F_n}) = h_μ(X, G),where G_μ is the set of generic points for μ with respect to {F_n} and h_top~B(G_μ, {F_n}) is the Bowen topological entropy(along {F_n}) on G_μ. This generalizes the classical result of Bowen(1973).     

Designs associated with maximum independent sets of a graph

A (v, β _o , μ)-design over regular graph G = (V, E) of degree d is an ordered pair D = (V, B), where |V| = v and B is the set of maximum independent sets of G called blocks such that if i, j ∈ V, i ≠ j and if i and j are not adjacent in G then there are exactly μ blocks containing i and j. In this paper, we study (v, β _o , μ)-designs over the graphs K _n × K _n , T(n)-triangular graphs, L ₂(n)-square lattice graphs, Petersen graph, Shrikhande graph, Clebsch graph and the Schläfli graph and non-existence of (v, β _o , μ)-designs over the three Chang graphs T ₁(8), T ₂(8) and T ₃(8).     

A note on K<Emphasis Type="Italic">ä</Emphasis>hler manifolds with almost nonnegative bisectional curvature

In this note, we prove the following result. There is a positive constant ε(n, Λ) such that if M ⁿ is a simply connected compact Kähler manifold with sectional curvature bounded from above by Λ, diameter bounded from above by 1, and with holomorphic bisectional curvature H ≥ ?ε(n, Λ), then M ⁿ is diffeomorphic to the product M ₁ × ? × M _k , where each M _i is either a complex projective space or an irreducible Kähler–Hermitian symmetric space of rank ≥ 2. This resolves a conjecture of Fang under the additional upper bound restrictions on sectional curvature and diameter.