Hardy spaces for the strip

In this paper we shall study Hardy spaces of analytic functions in a strip S. Our main result is on one hand an intrinsic characterization of the spaces and on the second that polynomials are dense. We also present an orthogonal (in H²(S)) basis of polynomials.     

Property C. Wallman basis and S-metrizability

In this present paper we prove that every Lindelof space which has a perfect locally connected Hausdorff compactification, has property C. (This latter concept was introduced by R.F. Dickman Jr). We make clear that this class of Lindelöf spaces properly contains the class of paracompact, connected, locally compact and locally connected spaces, as well as the class of those spaces whose topology can be induced by a metric with property S (or S-metrizable spaces). In this fashion, we simultaneously generalize two previous results of Dickman on spaces with property C. The use of Wallman basis with certain connectedness properties turns out to be a very convenient tool in the construction of locally connected compactifications as well as in characterizing S-metrizable spaces.     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.     

A cooperative location game based on the 1-center location problem

In this paper we introduce and analyze new classes of cooperative games related to facility location models defined on general metric spaces. The players are the customers (demand points) in the location problem and the characteristic value of a coalition is the cost of serving its members. Specifically, the cost in our games is the service radius of the coalition. We call these games the Minimum Radius Location Games (MRLG).We study the existence of core allocations and the existence of polynomial representations of the cores of these games, focusing on network spaces, i.e., finite metric spaces induced by undirected graphs and positive edge lengths, and on the ?_p metric spaces defined over R^d.     

Reversibility properties of semigroup actions on homogeneous spaces

This paper studies reversibility of subsemigroups acting on homogeneous spaces. The reversor set of a subsemigroup is defined and it is related to the invariant control sets for semigroups acting on certain homogeneous spaces. Let G be a connected noncompact semi-simple Lie group with finite center. Let L be a subgroup of?G. Assume that S is a subsemigroup of G with intS???. The main result characterizes the reversibility of the S-action on G/L in terms of the actions of S and L on the flag manifolds of G.     

Cyclic gradings of Lie algebras and lax pairs for <Emphasis Type="Italic">σ</Emphasis>-models

We study a class of σ-models with complex homogeneous target spaces and zero-curvature representations. We find a relation between these models and σ-models with certain m-symmetric target spaces. We also describe a model with the hypercomplex target space S ¹ × S ³ in detail.     

The weak McShane integral

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ-finite outer regular quasi Radon measure space (S,Σ, T, µ) into a Banach space X and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function f from S into X is weakly McShane integrable on each measurable subset of S if and only if it is Pettis and weakly McShane integrable on S. On the other hand, we prove that if an X-valued function is weakly McShane integrable on S, then it is Pettis integrable on each member of an increasing sequence (S _l ) _l?1 of measurable sets of finite measure with union S. For weakly sequentially complete spaces or for spaces that do not contain a copy of c ₀, a weakly McShane integrable function on S is always Pettis integrable. A class of functions that are weakly McShane integrable on S but not Pettis integrable is included.     

Linear homeomorphisms of spaces of continuous functions on long Sorgenfrey lines

We carry out a linear homeomorphic classification of the spaces of continuous functions on the long Sorgenfrey lines S_α, where a is an arbitrary ordinal. The spaces of continuous functions are endowed with the topology of pointwise convergence and denoted by C_p(S_α).     

S(n)-θ-closed spaces

A P-space is called P-θ-closed if it is θ-closed in every P-space in which it is embedded. S(n)-θ-closed spaces are characterized in terms of special covers and filters, and a categorical property of S(n)-θ-closed spaces is established. Examples are given to distinguish S(n)-θ-closed and S(n)-closed spaces and to show that the property of being S(n)-θ-closed is not productive.     

Weighted Hardy and singular operators in Morrey spaces

We study the weighted boundedness of the Cauchy singular integral operator SΓ in Morrey spaces Lp,λ(Γ) on curves satisfying the arc-chord condition, for a class of “radial type” almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces Lp,λ(0,?), ?>0. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves.     

The Sylvester-Gallai theorem, colourings and algebra

Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem:

LetSbe a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct pointsA,B∈Ssharing a colour there is a third pointC∈S, of the other colour, collinear withAandB. Then all the points inSare collinear.

We define a chromatic geometry to be a simple matroid for which each point is coloured red or blue or with both colours, such that for any two distinct points A,B∈S sharing a colour there is a third point C∈S, of the other colour, collinear with A and B. This is a common generalisation of proper finite linear spaces and properly two-coloured finite linear spaces, with many known properties of both generalising as well. One such property is Kelly’s complex Sylvester-Gallai theorem. We also consider embeddings of chromatic geometries in Desarguesian projective spaces. We prove a lower bound of 51 for the number of points in a three-dimensional chromatic geometry in projective space over the quaternions. Finally, we suggest an elementary approach to the corollary of an inequality of Hirzebruch used by Kelly in his proof of the complex Sylvester-Gallai theorem.     

Explicit characterizations of orthogonality in lS(C)

In a previous paper, the author used a notion of orthogonality introduced in another article to establish characterizations for orthogonality in the spaces l_S^p(C), 1?p<∞, thus obtaining generalizations of the usual characterization of orthogonality in the Hilbert spaces l_S²(C), via inner products. In this paper we make explicit these characterizations for some of the spaces l_S^p(C). We finish by presenting some remarks and open problems.     

Calabi's diastasis function for Hermitian symmetric spaces

In this paper we study the Calabi diastasis function of Hermitian symmetric spaces. This allows us to prove that if a complete Hermitian locally symmetric space (M,g) admits a Kähler immersion into a globally symmetric space (S,G) then it is globally symmetric and the immersion is injective. Moreover, if (S,G) is symmetric of a specified type (Euclidean, noncompact, compact), then (M,g) is of the same type. We also give a characterization of Hermitian globally symmetric spaces in terms of their diastasis function. Finally, we apply our analysis to study the balanced metrics, introduced by Donaldson, in the case of locally Hermitian symmetric spaces.     

On harmonic and 2-stein spaces of Iwasawa type

We study geometric properties of solvable metric Lie groups S of Iwasawa type; in particular harmonicity and the 2-stein condition. One restriction we obtain is that harmonic spaces of Iwasawa type have algebraic rank one, that is, the commutator subgroup of S has codimension one.We show that among Carnot solvmanifolds the only harmonic spaces are the Damek–Ricci spaces. Moreover, this rigidity result remains valid if harmonicity is replaced by the weaker 2-stein condition. As an application, we show that a harmonic Lie group of Iwasawa type with nonsingular 2-step nilpotent commutator subgroup is, up to scaling, a Damek–Ricci space.     

On powers of certain lines

In this paper we answer questions of van Douwen and Pfeffer by showing that the spaces S, S², S³, and T, T², T³,... are topologically distinct, where S is the Sorgenfrey line and T is the set of irrational points in S. We obtain analogous results for the Michael line M and present related examples.     

Criteria for the compactness and the Fredholm property of pseudodifferential operators in Hölder-Zygmund weighted spaces

We obtain necessary and sufficient conditions for the complete continuity (the Fredholm property) in Hölder-Zygmund spaces on ? ⁿ whose weight has a power-law behavior at infinity for pseudodifferential operators with symbols in the Hörmander class S _1,δ ^m , 0 ≤ δ < 1 (slowly varying symbols in the class S _1,0 ^m ). We show that such operators are compact operators or Fredholm operators in weighted Hölder-Zygmund spaces if and only if they are compact operators or Fredholm operators, respectively, in Sobolev spaces.     

Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P².The generic noncommutative plane corresponds to the Sklyanin algebra S=Skl(E,σ) constructed from an automorphism σ of infinite order on an elliptic curve E⊂P². In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1−n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P²?E.     

Decomposition of Besov and Triebel-Lizorkin spaces on the sphere

A discrete system of almost exponentially localized elements (needlets) on the n-dimensional unit sphere Sn is constructed. It shown that the needlet system can be used for decomposition of Besov and Triebel-Lizorkin spaces on the sphere. As an application of Besov spaces on Sn, a Jackson estimate for nonlinear m-term approximation from the needlet system is obtained.     

Ramsey property, ultrametric spaces, finite posets, and universal minimal flows

We introduce the class COU _S of finite ultrametric spaces with distances in the set S and with two additional linear orderings. We also introduce the class EOP of finite posets with two additional linear orderings. In this paper, we prove that COU _S and EOP are Ramsey classes. In addition, we give an application of our results to calculus of universal minimal flows.