Qp Spaces Research Articles

Cardinal invariants and special maps of quasicontinuous functions with the topology of pointwise convergence

For topological spaces X and Y, let Qp(X,Y) be the space of all quasicontinuous functions from X to Y with the topology of pointwise convergence. In this paper, we study the cardinal invariants such as cellularity, character, weight, density, pseudocharacter and spread of the space Qp(X,Y). We also discuss the properties of the restriction and induced maps related to the space Qp(X,Y).

A universal coregular countable second-countable space

A Hausdorff topological space X is called superconnected (resp. coregular) if for any nonempty open sets U1,…Un⊆X, the intersection of their closures U‾1∩…∩U‾n is not empty (resp. the complement X∖(U‾1∩…∩U‾n) is a regular topological space). A canonical example of a coregular superconnected space is the projective space QP∞ of the topological vector space Q<ω={(xn)n∈ω∈Qω:|{n∈ω:xn≠0}|<ω} over the field of rationals Q. The space QP∞ is the quotient space of Q<ω∖{0}ω by the equivalence relation x∼y iff Q⋅x=Q⋅y.We prove that every countable second-countable coregular space is homeomorphic to a subspace of QP∞, and a topological space X is homeomorphic to QP∞ if and only if X is countable, second-countable, and admits a decreasing sequence of closed sets (Xn)n∈ω such that (i) X0=X, ⋂n∈ωXn=∅, (ii) for every n∈ω and a nonempty relatively open set U⊆Xn the closure U‾ contains some set Xm, and (iii) for every n∈ω the complement X∖Xn is a regular topological space. Using this topological characterization of QP∞ we find topological copies of the space QP∞ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.

Qp spaces for weighted Möbius actions

We extend the classical theory of Qp-spaces of Aulaskari, Xiao and Zhao in two directions: first, by considering more general construction of such spaces; and second, by considering invariance with respect to weighted Möbius actions. In addition, related results for the associated Bloch-type spaces are also established.

Möbius invariant Dirichlet type spaces

Let Dμ,p be a Dirichlet type space induced by a positive parameter p and a positive Borel measure μ on the open unit disk. Denote by M(Dμ,p) the Möbius invariant function space generated by Dμ,p. It is known that if the measure μ is finite, then M(Dμ,p) is equal to the well-known Möbius invariant space Qp. In this paper, we investigate Dμ,p and M(Dμ,p) spaces when the measures μ are not necessarily finite. We give the relation between M(Dμ,p) and the Bloch space. We characterize inner functions in M(Dμ,p) spaces. We also consider a Carleson measure problem for Dμ,p spaces.

On Classes for Hyperbolic Riemann Surfaces

AbstractThe Qpspaces of holomorphic functions on the disk, hyperbolic Riemann surfaces or complex unit ball have been studied deeply. Meanwhile, there are a lot of papers devoted to theclasses of meromorphic functions on the disk or hyperbolic Riemann surfaces. In this paper, we prove the nesting property (inclusion relations) ofclasses on hyperbolic Riemann surfaces. The same property for Qp spaces was also established systematically and precisely in earlier work by the authors of this paper.

Jackson's Theorem in Qp Spaces in Polydiscs

Jackson's theorem is an important result in the theory of approximation of function. The purpose of this article is to introduce a new kind of Qp spaces in the unit polydiscs. Jackson's approximation theorem is established in Qp spaces.

Connections Of Linear Operators Defined By Analytic Functions With Qp Spaces

This paper is concerned mainly with the linear operators \(I_f^{\gamma , \alpha}\) and \(J_f^{\gamma , \alpha}\) of analytic function \(f\).The norm of \(I_f^{\gamma , \alpha}\) and \(J_f^{\gamma , \alpha}\) on some analytic function spaces is computed in this paper. We study the relation between \(I_f^{\gamma , \alpha}\) and \(J_f^{\gamma , \alpha}\) operators, the \(\beta(\lambda)\) spaces and \(Q_p\) spaces \((0<p<\infty)\).

Ricci tensor of slant submanifolds in a quaternion projective space

We study slant submanifolds of quaternion Kaehler manifold and in particular of a quaternion projective space. We obtain a sharp estimate of the Ricci tensor of a slant submanifold M in a quaternion projective space QP m ( 4 c ) in terms of the main extrinsic invariant, namely the square mean curvature.

Scalar curvature of QR-submanifolds with maximal QR-dimension in a quaternionic projective space

In this paper we derive an integral formula on an n-dimensional, compact, minimal QR-submanifoldM of (p−1) QR-dimension immersed in a quaternionic projective space QP(n+p)/4. Using this integral formula, we give a sufficient condition concerning with the scalar curvature of M in order that such a submanifold M is to be a tube over a quaternionic projective space.

Riemann–Stieltjes operators and multipliers on [formula omitted] spaces in the unit ball of [formula omitted

This paper is devoted to characterizing the Riemann–Stieltjes operators and pointwise multipliers acting on Möbius invariant spaces Qp, which unify BMOA and Bloch space in the scale of p. The boundedness and compactness of these operators on Qp spaces are determined by means of an embedding theorem, i.e. Qp spaces boundedly embedded in the non-isotropic tent type spaces Tq∞.

Blaschke products, α-curves and Qp spaces

AbstractWe give necessary and sufficient conditions for Blaschke products with zeros on an

Carleson measures and the BMO space on the p ‐adic vector space

AbstractFor a prime number p, let Qp be the p ‐adic field and let Qp d denote a vector space over Qp which consists of all d ‐tuples of Qp . Then we study the p ‐adic version of the Calderón–Zygmund decomposition, Carleson measures on the vector space Qp d +1 and the space BMO (Qp d ) of functions of bounded mean oscillation on Qp d . In particular, it turns out that the operator norms of various oncoming operators are independent of the dimension d and the prime number p, which is one of the big differences from that of the Euclidean case. Interestingly, the independence of the dimension d and p makes it possible to develop Harmonic Analysis on the infinite dimensional p ‐adic vector space as the importance had already been pointed out in the Euclidean case (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

First passage time distribution and the number of returns for ultrametric random walks

In this paper, we consider a homogeneous Markov process ξ(t; ω) on an ultrametric space Qp, with distribution density f(x, t), x ∊ Qp, t ∊ R+, satisfying the equation , usually called the ultrametric diffusion equation. We construct and examine a random variable that has the meaning the first passage times. Also, we obtain a formula for the mean number of returns on the interval (0, t] and give its asymptotic estimates for large t.

Qp-valued jump processes associated with linear and nonlinear pseudo-differential equations

We consider stochastic differential equations in the space Qp of p-adic numbers associated with nonlinear pseudo-differential equations and systems. We construct both scalar and matrix-valued multiplicative functionals of Qp-valued Markov processes and derive probabilistic representations of solutions to the Cauchy problem for nonlinear pseudo-differential equations and systems which extend the Feynman-Kac formula. Bibliography: 12 titles.

$Q_K$ Spaces Via Higher Order Derivatives

We characterize the Mobius invariant QK spaces in terms of higher order derivatives. Our methods are new even in the case of Qp spaces.

Qp‐spaces on bounded symmetric domains

We generalize the theory of Qp spaces, introduced on the unit disc in 1995 by Aulaskari, Xiao and Zhao, to bounded symmetric domains in Cd, as well as to analogous Moebius‐invariant function spaces and Bloch spaces defined using higher order derivatives; the latter generalization contains new results even in the original context of the unit disc.

Preduals of Qp -spaces II. Carleson imbeddings and atomic decompositions

In [Aleman, A., Carlsson, M. and Persson A., Preduals of Qp -spaces. Complex Variables (To appear).], we have obtained a representation of the Cauchy-predual of the space Qp on the unit disc as a weak product of weighted Dirichlet and Bergman spaces. The present article is a continuation of Aleman et al. and contains several applications and further developments of those results. We investigate the relation between Qp and Carleson inequalities for functions in weighted Dirichlet spaces and, in particular, we prove a characterization of bounded Qp -functions in terms of pointwise multipliers between such spaces. Moreover, we use our approach based on imbeddings in vector-valued sequence spaces to obtain atomic decompositions of the predual of Qp . This last result is then extended to the real variable setting where we prove atomic decomposition theorems for the preduals of certain function spaces that generalize .

Preduals of Qp -spaces

We prove a weak factorization result for the predual of the space Qp on the unit disc, that is, we show that functions in this space can be written as sums of products of functions in given weighted Dirichlet and Bergman spaces with the usual control on the norms.

Sectional curvature of QR-submanifolds of (p-1) QR-dimension in a quaternionic projective space QP^{(n+p)/4}

Sectional curvature of QR-submanifolds of (p-1) QR-dimension in a quaternionic projective space QP^{(n+p)/4}

Randomization of Qp spaces on the unit ball of ℂn

In this paper, we study the random power series of several complex variables, and give some sufficient conditions for random power series to belong to Q(p) and Q(p,0) spaces by means of their characterizations of Carleson measure. Moreover, we show that such conditions are best possible in some sense.