Continuous embeddings in harmonic mixed norm spaces on the unit ball in ? n

In this paper continuous embeddings in spaces of harmonic functions with mixed norm on the unit ball in ? ⁿ are established, generalizing some Hardy-Littlewood embeddings for similar spaces of holomorphic functions in the unit disc. Differences in indices between the spaces of harmonic and holomorphic spaces are revealed. As a consequence an analogue of classical Fejér-Riesz inequality is obtained. Embeddings in the special case of Riesz systems are also established.     

Area operator on Bergman spaces

We characterize the non-negative measuresμon the unit disk D for which the area operator Aμis bounded from Bergman space Aαp to Lq ((?)D).     

Bergman type operators on mixed norm spaces over the ball in ℂ<Superscript>n</Superscript>

The paper considers Bergman type operators introduced by Shields and Williams depending on a normal pair of weight functions. We prove that there exist values of parameter ß for which these operators are bounded on mixed norm spaces L(p, q, ß) on the unit ball in Cⁿ.     

Randomization of Q_p spaces on the unit ball of C~n

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 Qp and Qp,0 spaces by means of their characterizations of Carleson measure. Moreover, we show that such conditions are best possible in some sense.     

Transitivity of the norm on Banach spaces¶ having a Jordan structure

We study transitivity conditions on the norm of JB ^*-triples, C ^*-algebras, JB-algebras, and their preduals. We show that, for the predual X of a JBW ^*-triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The closed unit ball of X has some extreme point and the norm of X is convex transitive. ii) The set of all extreme points of the closed unit ball of X is non rare in the unit sphere of X. These results are applied to obtain partial affirmative answers to the open problem whether every JB ^*-triple with transitive norm is a Hilbert space. We extend to arbitrary C ^*-algebras previously known characterizations of transitivity [20] and convex transitivity [36] of the norm on commutative C ^*-algebras. Moreover, we prove that the Calkin algebra has convex transitive norm. We also prove that, if X is a JB-algebra, and if either the norm of X is convex transitive or X has a predual with convex transitive norm, then X is associative. As a consequence, a JB-algebra with almost transitive norm is isomorphic to the field of real numbers. Received: 9 June 1999 / Revised version: 20 February 2000     

Explicit min–max polynomials on the disc

Denote by ${\Pi }_{n+m?1}^2?\{{\sum }_{0\le i+j\le n+m?1}{c}_{i,j}{x}^i{y}^j:c_{i,j}\in \mathbb{R}\}$ the space of polynomials of two variables with real coefficients of total degree less than or equal to $n+m?1$. Let $b_0,b_1,\dots ,b_l\in \mathbb{R}$ be given. For $n,m\in \mathbb{N},n\ge l+1$ we look for the polynomial $b_0{x}^n{y}^m+b_1{x}^{n?1}{y}^{m+1}+?+b_l{x}^{n?l}{y}^{m+l}+q(x,y),q(x,y)\in {\Pi }_{n+m?1}^2$, which has least maximum norm on the disc and call such a polynomial a min–max polynomial. First we introduce the polynomial $2P_{n,m}(x,y)=x{G}_{n?1,m}(x,y)+y{G}_{n,m?1}(x,y)=2x^n{y}^m+q(x,y)$ and $q(x,y)\in {\Pi }_{n+m?1}^2$, where $G_{n,m}(x,y)?1/2^{n+m}(U_n\left(x\right)U_m\left(y\right)+U_{n?2}\left(x\right)U_{m?2}\left(y\right))$, and show that it is a min–max polynomial on the disc. Then we give a sufficient condition on the coefficients $b_j,j=0,\dots ,l,l$ fixed, such that for every $n,m\in \mathbb{N},n\ge l+1$, the linear combination ${\sum }_{\nu =0}^l{b}_{\nu }{P}_{n?\nu ,m+\nu }(x,y)$ is a min–max polynomial. In fact the more general case, when the coefficients $b_j$ and $l$ are allowed to depend on $n$ and $m$, is considered. So far, up to very special cases, min–max polynomials are known only for $x^n{y}^m$,$n,m\in {\mathbb{N}}_0$.     

Weighted composition operators between μ-Bloch spaces on the unit ball

In this paper,necessary and sufficient conditions are given for the weighted composition operator T_(ψ,ψ) to be bounded or compact from the space β_μto β_ν(or β_(μ,0) to β_(ν,0) ) on the unit ball of C~n.At the same time,a series of corollaries are also obtained.     

Weighted composition operators between μ -bloch spaces on the unit ball

In this paper, necessary and sufficient conditions are given for the weighted composition operator T _ψ,φ to be bounded or compact from the space β_μ to β_υ (or β_μ,0 to β _υ,0)on the unit ball of C ^rn. At the same time, a series of corollaries are also obtained.     

Multiplication operator on the weighted Bergman space of the unit ball北大核心CSCD

Multiplication operator is an important class of operators in function space. We mainly research the properties of the multiplication operator on the weighted Bergman space of the unit ball. ©, 2015, Chinese Academy of Sciences. All right reserved.     

On the convexity coefficient of Musielak–Orlicz function spaces equipped with the Orlicz norm

In Hudzik and Landes, the convexity coefficient of Musielak–Orlicz function spaces over a non-atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak–Orlicz functions are strictly convex see [6]. In this paper, we extend this result to the case of Musielak–Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak–Orlicz function spaces as well as k-uniformly convex Musielak–Orlicz spaces equipped with the Orlicz norm is given.     

Integral-type operators between α-Bloch spaces and Besov spaces on the unit ball

Let H(B) denote the space of all holomorphic functions on the unit ball B⊂Cⁿ. The boundedness and compactness of the following integral-type operators

     

Composition Operators on Weighted Bergman–Orlicz Spaces on the Ball

We give embedding theorems for weighted Bergman–Orlicz spaces on the ball and then apply our results to the study of the boundedness and the compactness of composition operators in this context. As one of the motivations of this work, we show that there exist some weighted Bergman–Orlicz spaces, different from H ^∞, on which every composition operator is bounded.     

The Nevanlinna–Pick problem on the closed unit disk: Minimal norm rational solutions of low degree

For the Nevanlinna–Pick interpolation problem with n$n$ interpolation conditions (interior and boundary), we construct a family of rational solutions of degree at most n−1$n-1$. We also establish necessary and sufficient conditions for the existence and the uniqueness of a solution with the minimally possible H^∞$H^{\infty }$-norm and construct a family of minimal-norm rational solutions of degree at most n−1$n-1$ in the indeterminate case. Finally, we supplement a result of Ruscheweyh and Jones showing that in case the interpolation nodes and the target values are all unimodular, any rational solution of degree at most n−1$n-1$ is necessarily a finite Blaschke product.     

The Gleason''''s problem on mixed norm spaces in convex domains

Let Ω be a bounded convex domain with C2 boundary in C2 and for given 0 < p, q ≤∞ and normal weight function (r) let Hp,q, be the mixed norm space on Ω. In this paper we prove that the Gleason's problem (Ω, a, Hp,q,) is solvable for any fixed point a ∈ Ω. While solving the Gleason's problem we obtain the boundedness of certain integral operator on Hp,q,.     

Oscillation of holomorphic Bergman–Besov kernels on the ball

We estimate the oscillation of holomorphic Bergman–Besov reproducing kernels on the unit ball of \(\mathbb {C}^n\). As an application of this estimate we characterize holomorphic Bergman–Besov spaces \(A_\alpha ^p\,(\alpha \in \mathbb {R})\) in terms of double integrals of the fractions \(|f(z)-f(w)|/|z-w|\) and \(|f(z)-f(w)|/|1-\langle z,w \rangle |\) and complete the earlier works done on this subject. Our results provide, when \(\alpha \le -1\), a derivative-free characterization of \(A_\alpha ^p\).     

Fejér–Riesz type inequalities for Bergman spaces

We obtain Fejér?CRiesz type inequalities for the weighted Bergman spaces on the unit disk of the complex plane. We show that the Fejér?CRiesz inequalities can be expressed as boundedness and compactness problems for certain Toeplitz operators.     

A remark on the Bergman kernels of the Cartan–Hartogs domains

We give a new formula for the Bergman kernels of the Cartan–Hartogs domains. As an application of our formula, we study the Lu Qi-Keng problem of the Cartan–Hartogs domains.