On the extent of star countable spaces

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ? X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω ₁-monolithic compact space X, if C _p (X)is star countable then it is Lindelöf.     

Interpolation of vector measures

Let (Ω, Σ) be a measurable space and m ₀: Σ → X ₀ and m ₁: Σ → X ₁ be positive vector measures with values in the Banach Köthe function spaces X ₀ and X ₁. If 0 < α < 1, we define a new vector measure [m ₀, m ₁] _α with values in the Calderón lattice interpolation space X ₀ ^1?ga X ₁ ^α and we analyze the space of integrable functions with respect to measure [m ₀, m ₁] _α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of L ^p -spaces. Since each p-convex order continuous Köthe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.     

Extrapolation in Grand Lebesgue Spaces with <Emphasis Type="Italic">A</Emphasis>∞ Weights

Results on extrapolation withA∞ weights in grand Lebesgue spaces are obtained. Generally, these spaces are defined with respect to the productmeasure μ₁ ×· · ·×μ_n onX₁ ×· · ·×X_n, where (X_i, d_i, μ_i), i = 1,..., n, are spaces of homogeneous type. As applications of the obtained results, new one-weight estimates with A_∞ weights for operators of harmonic analysis are derived.     

Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L ^p and weak L ^p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L _Φ having the property L ^∞ ? L _Φ L ^p , 1 ? p > ∞. The second contains spaces L _Φ that resemble L ^p spaces.     

Theory of (<Emphasis Type="Italic">q</Emphasis> <Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis> <Subscript>2</Subscript>)-quasimetric spaces and coincidence points

We introduce (q₁, q₂)-quasimetric spaces and examine their properties. Covering mappings between (q₁, q₂)-quasimetric spaces are investigated. Sufficient conditions for the existence of a coincidence point of two mappings acting between (q₁, q₂)-quasimetric spaces such that one is a covering mapping and the other satisfies the Lipschitz condition are obtained.     

The Fixed Points of Contractions of <Emphasis Type="Italic">f</Emphasis>-Quasimetric Spaces

The recent articles of Arutyunov and Greshnov extend the Banach and Hadler Fixed-Point Theorems and the Arutyunov Coincidence-Point Theorem to the mappings of (q₁, q₂)-quasimetric spaces. This article addresses similar questions for f-quasimetric spaces.Given a function f: R ₊² → R₊ with f(r₁, r₂) → 0 as (r₁, r₂) → (0, 0), an f-quasimetric space is a nonempty set X with a possibly asymmetric distance function ρ: X² → R+ satisfying the f-triangle inequality: ρ(x, z) ≤ f(ρ(x, y), ρ(y, z)) for x, y, z ∈ X. We extend the Banach Contraction Mapping Principle, as well as Krasnoselskii’s and Browder’s Theorems on generalized contractions, to mappings of f-quasimetric spaces.     

Decomposition theorems for<Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> spaces

We study the Möbius invariant spacesQ _p andQ _{p, 0} of analytic functions. These scales of spaces include BMOA=Q₁, VMOA=Q_{1, 0} and the Dirichlet space=Q₀. Using the Bergman metric, we establish decomposition theorems for these spaces. We obtain also a fractional derivative characterization for bothQ _p andQ _{p, 0} .     

Some specific unboundedness property in smoothness Morrey spaces. The non-existence of growth envelopes in the subcritical case

We study smoothness spaces of Morrey type on Rn and characterise in detail those situations when such spaces of type A_(p,q)~(s,r)(R~n) or A_(u,p,q)~s(R~n) are not embedded into L_(∞)(R~n).We can show that in the so-called sub-critical,proper Morrey case their growth envelope function is always infinite which is a much stronger assertion.The same applies for the Morrey spaces M_(u,p)(R~m) with p u.This is the first result in this direction and essentially contributes to a better understanding of the structure of the above spaces.     

Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups

In this paper, we study the boundedness of the fractional integral operator I _α on Carnot group G in the generalized Morrey spaces M _{p, φ} (G). We shall give a characterization for the strong and weak type boundedness of I _α on the generalized Morrey spaces, respectively. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev–Stein embedding theorems on generalized Morrey spaces in the Carnot group setting.     

Vectorial Ekeland variational principle for systems of equilibrium problems and its applications

For a family of vector-valued bifunctions,we introduce the notion of sequentially lower monotonity,which is strictly weaker than the lower semi-continuity of the second variables of the bifunctions.Then,we give a general version of vectorial Ekeland variational principle(briefly,denoted by EVP) for a system of equilibrium problems,where the sequentially lower monotone objective bifunction family is defined on products of sequentially lower complete spaces(concerning sequentially lower complete spaces,see Zhu et al(2013)),and taking values in a quasi-ordered locally convex space.Besides,the perturbation consists of a subset of the ordering cone and a family {p_i}_(i∈I) of negative functions satisfying for each i∈I,p_i(x_i,y_i) = 0 if and only if x_i=y_i.From the general version,we can deduce several particular equilibrium versions of EVP,which can be applied to show the existence of solutions for countable systems of equilibrium problems.In particular,we obtain a general existence result of solutions for countable systems of equilibrium problems in the setting of sequentially lower complete spaces.By weakening the compactness of domains and the lower semi-continuity of objective bifunctions,we extend and improve some known existence results of solutions for countable system of equilibrium problems in the setting of complete metric spaces(or Fréchet spaces).When the domains are non-compact,by using the theory of angelic spaces(see Floret(1980)),we generalize some existence results on solutions for countable systems of equilibrium problems by extending the framework from reflexive Banach spaces to the strong duals of weakly compactly generated spaces.     

A Note on Campanato Spaces and Their Applications

In this paper, we obtain a version of the John–Nirenberg inequality suitable for Campanato spaces C_p,β with 0 < p < 1 and show that the spaces C_p,β are independent of the scale p ∈ (0,∞) in sense of norm when 0 < β < 1. As an application, we characterize these spaces by the boundedness of the commutators [b,B _α ] _j (j = 1, 2) generated by bilinear fractional integral operators B _α and the symbol b acting from L_p1 × L_p2 to L ^q for p1, p2 ∈ (1,∞), q ∈ (0,∞) and 1/q = 1/p1 + 1/p2 ? (α + β)/n.     

New characterizations for weighted composition operator from Zygmund type spaces to Bloch type spaces

Let u be a holomorphic function and φ a holomorphic self-map of the open unit disk \(\mathbb{D}\) in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators uC _φ from Zygmund type spaces to Bloch type spaces in \(\mathbb{D}\) in terms of u, φ, their derivatives, and φ ⁿ , the n-th power of φ. Moreover, we obtain some similar estimates for the essential norms of the operators uC _φ , from which sufficient and necessary conditions of compactness of uC _φ follows immediately.     

Frames and non linear approximations in Hilbert spaces

A simple proof of the proposition, stated in ([2], p. 346), asserting that in Hilbert spaces a Riesz basis is greedy, is given. Also, greedy approximant for frames in Hilbert spaces is defined and it is shown that frames satisfy the quasi greedy and almost greedy conditions. Finally, we give the characterizations of approximation spaces A^s(Ψ), A_q^s(Ψ) by means of weak-l_p and Lorentz sequence spaces for frames.     

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.     

On Symmetric Spaces With Convergence in Measure on Reflexive Subspaces

A closed subspace H of a symmetric space X on [0, 1] is said to be strongly embedded in X if in H the convergence in X-norm is equivalent to the convergence in measure. We study symmetric spaces X with the property that all their reflexive subspaces are strongly embedded in X. We prove that it is the case for all spaces, which satisfy an analogue of the classical Dunford–Pettis theorem on relatively weakly compact subsets in L₁. At the same time the converse assertion fails for a broad class of separableMarcinkiewicz spaces.     

Functor <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">τ</Emphasis></Subscript>, Lipschitzian and uniformly continuous mappings

The paper is devoted to questions on lifting of the functor M _τ : Tych → Tych to the categories of metric and uniform spaces. Similar problems were solved for the functor U _τ of the unit ball of τ-additive measures. The main difference between the functor M _τ and the functor U _τ is that the space M _τ (X) is compact only for X = Ø. A more delicate distinction is expressed by Theorem 2 which implies that the functor M _τ does not always preserve the uniform continuity of mappings of metric spaces (even in the case of compacta). Nevertheless, the problem of lifting the functor M _τ to the category Unif turns to be solvable.     

Packing,counting and covering Hamilton cycles in random directed graphs

In this paper we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian on weighted L ^p spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces L ^p (R ⁿ , (φ _iρ (x))² dx) where φ _iρ is the Euclidean spherical function. The behavior is very similar to the case of the Laplace–Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.     

Estimates of some integral operators with bounded variable kernels on the hardy and weak hardy spaces over ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we first introduce \({L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}\) conditions satisfied by the variable kernels Ω(x, z) for 0 ≤ σ ₁ ≤ 1 and σ ₂ ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators T _Ω, fractional integrals T _Ω,α and parametric Marcinkiewicz integrals μ _Ω ^ρ with variable kernels on the Hardy spaces H ^p (R ⁿ ) and weak Hardy spaces WH ^p (R ⁿ ). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy–Lorentz spaces H ^p,q(R ⁿ ) for all p < q < ∞.