Q spaces and Morrey spaces

We characterize functions in Morrey space by p-Carleson measures. We then reveal a simple relation between Q space and Morrey space, that is Q space can be viewed as a fractional integration of the Morrey space. Therefore, many results for Morrey space can be translated onto Q space. For example, we show that Q space is a dual space by identifying its predual.     

The relation between GTi-spaces and fuzzy proximity spaces, G-compact spaces, fuzzy uniform spaces

In this paper, we study the relation between the fuzzy separation axioms, which had been introduced by the authors in 2001, and the fuzzy proximity defined by Katsaras in 1980. We study also the relation between our fuzzy separation axioms and the G-compactness defined by Gähler in 1995. Moreover, we show here the relation between these fuzzy separation axioms and the fuzzy uniform structures introduced and studied by Gähler and the first author in 1998.     

W-Sobolev spaces

Fix strictly increasing right continuous functions with left limits and periodic increments, Wi:R→R, i=1,…,d, and let for x∈Rd. We construct the W-Sobolev spaces, which consist of functions f having weak generalized gradients ∇Wf=(W_1∂f,…,Wd_∂f). Several properties, that are analogous to classical results on Sobolev spaces, are obtained. Existence and uniqueness results for W-generalized elliptic equations, and uniqueness results for W-generalized parabolic equations are also established. Finally, an application of this theory to stochastic homogenization is presented.     

A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces

Let s∈R, τ∈[0,∞), p∈(1,∞) and q∈(1,∞]. In this paper, we introduce a new class of function spaces which unify and generalize the Triebel-Lizorkin spaces with both p∈(1,∞) and p=∞ and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the relationship between Triebel-Lizorkin spaces and Q spaces, which answers a question posed by Dafni and Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and Qα(Rn), J. Funct. Anal. 208 (2004) 377-422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces and determine their dual spaces , where s∈R, p,q∈[1,∞), max{p,q}>1, , and t^′ denotes the conjugate index of t∈(1,∞); as an application of this, we further introduce certain Hardy-Hausdorff spaces and prove that the dual space of is just when p,q∈(1,∞).     

On M-separability of countable spaces and function spaces

We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products, even for the spaces of continuous functions with the topology of pointwise convergence. We also show that there exists no maximal M-separable countable space in the model of Frankiewicz, Shelah, and Zbierski in which all closed P-subspaces of ω^* admit an uncountable family of nonempty open mutually disjoint subsets. This answers several questions of Bella, Bonanzinga, Matveev, and Tkachuk.     

On D-Paracompact spaces

Following Pareek a topological space X is called D-paracompact if for every open cover $\text{A}$ of X there exists a continuous mapping f from X onto a developable T₁-space Y and an open cover $\text{B}$ of Y such that { f^-1[B]|B ∈ $\text{B}$ } refines $\text{A}$. It is shown that a space is D-paracompact if and only if it is subparacompact and D-expandable. Moreover, it is proved that D-paracompactness coincides with a covering property, called dissectability, which was introduced by the author in order to obtain a base characterization of developable spaces.     

T 1 theorem for Besov spaces on nonhomogeneous spaces

Suppose μ is a Radon measure on ℝ ^d , which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀>0 such that for all x∈supp(μ) and r>0, μ(B(x, r))⪯C₀rⁿ, where 0<n⪯d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa’s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].     

Bounded tightness for locally convex spaces and spaces C(X)

We show that metrizability and bounded tightness are actually equivalent for a large class of locally convex spaces including (LF)-spaces, (DF)-spaces, the space of distributions D′(Ω), etc. A consequence of this fact is that for the bounded tightness for the weak topology of X is equivalent to the following one: X is linearly homeomorphic to a subspace of . This nicely supplements very recent results of Cascales and Raja. Moreover, we show that a metric space X is separable if the space C_p(X) has bounded tightness.     

Meet continuous lattices, limit spaces, and L-topological spaces

In this paper, a systematic investigation of the relationship between meet continuous lattices, limit spaces, and L-topological spaces is given. It is a continuation of the investigation on this topic by Höhle (2000, 2001). The relationship between the Lowen functors and the functors introduced by Höhle (2000, 2001) is made clear.     

Maps and f-normal spaces

It has long been known that hyper-real maps preserve realcompactness. In this paper we show that hyper-real maps preserve nearly realcompactness as well. We will also introduce the concepts of ε-perfect maps and f-normal spaces and explore them in a way that mirrors Rayburn's 1978 study of δ-perfect maps and h-normal spaces.     

On L-Tychonoff spaces II

This paper is a sequel to the 1995 paper On L-Tychonoff spaces. The embedding theorem for L-topological spaces is shown to hold true for L an arbitrary complete lattice without imposing any order reversing involution (·)^′ on L. Some results on completely L-regular spaces and on L-Tychonoff spaces, which have previously been known to hold true for (L,^′) a frame, are exhibited as ones holding for (L,^′) a meet-continuous lattice. For such a lattice an insertion theorem for completely L-regular spaces is given. Some weak forms of separating families of maps are discussed. We also clarify the dependence between the sub-T₀ separation axiom of Liu and the L-T₀ separation axiom of Rodabaugh.     

Efimov spaces and the separable quotient problem for spaces Cp(K)

The classic Rosenthal–Lacey theorem asserts that the Banach space $C(K)$ of continuous real-valued maps on an infinite compact space K has a quotient isomorphic to c or ${?}_2$. More recently, Ka?kol and Saxon [20] proved that the space $C_p(K)$ endowed with the pointwise topology has an infinite-dimensional separable quotient algebra iff K has an infinite countable closed subset. Hence $C_p(\beta \mathbb{N})$ lacks infinite-dimensional separable quotient algebras. This motivates the following question: (?) Does$C_p(K)$admit an infinite-dimensional separable quotient (shortly SQ) for any infinite compact space K? Particularly, does $C_p(\beta \mathbb{N})$ admit SQ? Our main theorem implies that $C_p(K)$ has SQ for any compact space K containing a copy of $\beta \mathbb{N}$. Consequently, this result reduces problem (?) to the case when K is an Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of $\beta \mathbb{N}$). Although, it is unknown if Efimov spaces exist in ZFC, we show, making use of some result of R. de la Vega (2008) (under ?), that for some Efimov space K the space $C_p(K)$ has SQ. Some applications of the main result are provided.     

INTERPOLATION SPACES BETWEEN H^1 AND L^∞ ON SPACES OF HOMOGENEOUS TYPE

Using the maximal function characterization of Hardy spaces,we study the interpolation spaces between H^1 and L^∞on spaces of homogeneous type.     

Topological ordered C- (resp. I-)spaces and generalized metric spaces

The following result due to Hanai, Morita, and Stone is well known: Let f be a closed continuous map of a metric space X onto a topological space Y. Then the following statements are equivalent: (i) Y satisfies the first countability axiom; (ii) for each y∈Y, f−1{y} has a compact boundary in X; (iii) Y is metrizable.In this article we obtain several related results in the setting of topological ordered spaces. In particular we investigate the upper and lower topologies of metrizable topological ordered spaces which are both C- and I-spaces in the sense of Priestley.     

Local n-connectivity of decomposition spaces

A corollary of the main result of this paper is the following Theorem. Suppose f: X → Y is a closed surjection of metrizable spaces whose point inverses are LC^{n + 1}-divisors (n ? 1). If Y is complete and f is homology n-stable, then Y is LC^{n + 1}provided X is LC^{n + 1}.Intuitively, f is homology n-stable if the ?ech homology groups of its point inverses are locally constant up to dimension n. In addition, we obtain sufficient conditions for the Freudenthal compactification to be LCⁿ.