Unit Ball Bn Research Articles

Composition operators on strictly pseudoconvex domains with smooth symbol

It is well known that the composition operator C is unbounded on Hardy and Bergman spaces on the unit ball Bn in C n when n> 1 for a linear holomorphic self-map of Bn. We find a sufficient and necessary condition for a composition operator with smooth symbol to be bounded on Hardy or Bergman spaces over a bounded strictly pseudoconvex domain in C n . Moreover, we show that this condition is equivalent to the compactness of the composition operator from a Hardy or Bergman space into the Bergman space whose weight is 1 bigger. We also prove that a certain jump phenomenon occurs when the composition operator is not bounded. Our results generalize known results on the unit ball to strictly pseudoconvex domains.

Joint Carleson measure and the difference of composition operators on [formula omitted

We introduce a concept of joint Carleson measure and characterize when the difference of two composition operators on Aαp(Bn), the weighted Bergman space over the unit ball Bn in Cn, is bounded or compact. We apply this joint Carleson measure characterization to composition operators with smooth symbols and construct an interesting example which shows that the boundedness or the compactness depends on p when n≥2. This is in sharp contrast with the single composition operator case where the boundedness or the compactness is independent of p>0. Moreover, the compact difference on the weighted Bergman spaces over the unit disc is known to be independent of p>0, and the compact difference on Aαp(Bn) is known to be independent of p>0 if each composition operator is bounded on Aβp(Bn) for some −1<β<α[2].

Uniformly starlike mappings and uniformly convex mappings on the unit ball Bn

In this article, we extend the definition of uniformly starlike functions and uniformly convex functions on the unit disk to the unit ball in ▪n, give the discriminant criterions for them, and get some inequalities for them.

Composition Operators Between Bloch Type Spaces in the Unit Ball

In this paper, we obtain some new necessary and sufficient conditions for the boundedness and compactness of composition operators ϕ between Bloch type spaces in the unit ball Bn.

Composition operators from logarithmic Bloch spaces to weighted Bloch spaces

We characterize the analytic self-maps ϕ of the unit disk D in C that induce continuous composition operators Cϕ from the log-Bloch space Blog(D) to μ-Bloch spaces Bμ(D) in terms of the sequence of quotients of the μ-Bloch semi-norm of the nth power of ϕ and the log-Bloch semi-norm of the nth power Fn of the identity function on D, where μ:D→(0,∞) is continuous and bounded. We also obtain an expression that is equivalent to the essential norm of Cϕ between these spaces, thus characterizing ϕ such that Cϕ is compact. After finding a pairwise norm equivalent family of log-Bloch type spaces that are defined on the unit ball Bn of Cn and include the log-Bloch space, we obtain an extension of our boundedness/compactness/essential norm results for Cϕ acting on Blog to the case when Cϕ acts on these more general log-Bloch-type spaces.

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in ℂn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f(·, 0) of a Loewner chain f(z, t) = etz + … such that {e−tf(·, t)}t⩾0 is a normal family on Bn. We show that if f(·, 0) is an extreme point (respectively a support point) of S0(Bn), then e−tf(·, t) is an extreme point of S0(Bn) for t ⩽ 0 (respectively a support point of S0(Bn) for t ∈ [0,t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.

Properties of Toeplitz Operators on Some Holomorphic Banach Function Spaces

We characterize complex measures <svg style="vertical-align:-2.29482pt;width:9.6374998px;" id="M1" height="9.6750002" version="1.1" viewBox="0 0 9.6374998 9.6750002" width="9.6374998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.675)"> <g transform="translate(72,-64.26)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan> </text> </g> </g> </svg> on the unit ball of <svg style="vertical-align:-0.17555pt;width:17.200001px;" id="M2" height="11.625" version="1.1" viewBox="0 0 17.200001 11.625" width="17.200001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.625)"> <g transform="translate(72,-62.7)"> <text transform="matrix(1,0,0,-1,-71.95,62.92)"> <tspan style="font-size: 12.50px; " x="0" y="0">ℂ</tspan> </text> <text transform="matrix(1,0,0,-1,-63.14,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan> </text> </g> </g> </svg>, for which the general Toeplitz operator <svg style="vertical-align:-4.74141pt;width:18.012501px;" id="M3" height="17.112499" version="1.1" viewBox="0 0 18.012501 17.112499" width="18.012501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.1125)"> <g transform="translate(72,-58.31)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑇</tspan> </text> <text transform="matrix(1,0,0,-1,-63.07,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝛼</tspan> <tspan style="font-size: 8.75px; " x="-2" y="8.1300001">𝜇</tspan> </text> </g> </g> </svg> is bounded or compact on the analytic Besov spaces <svg style="vertical-align:-4.74141pt;width:44.012501px;" id="M4" height="16.6" version="1.1" viewBox="0 0 44.012501 16.6" width="44.012501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.6)"> <g transform="translate(72,-58.72)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐵</tspan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑝</tspan> </text> <text transform="matrix(1,0,0,-1,-58.47,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝔹</tspan> </text> <text transform="matrix(1,0,0,-1,-45.85,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan> </text> <text transform="matrix(1,0,0,-1,-41.01,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">)</tspan> </text> </g> </g> </svg>, also on the minimal M&#xf6;bius invariant Banach spaces <svg style="vertical-align:-3.20526pt;width:44.125px;" id="M5" height="14.6875" version="1.1" viewBox="0 0 44.125 14.6875" width="44.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6875)"> <g transform="translate(72,-60.25)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐵</tspan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">1</tspan> </text> <text transform="matrix(1,0,0,-1,-58.37,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝔹</tspan> </text> <text transform="matrix(1,0,0,-1,-45.76,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan> </text> <text transform="matrix(1,0,0,-1,-40.91,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">)</tspan> </text> </g> </g> </svg> in the unit ball <svg style="vertical-align:-3.20526pt;width:16.75px;" id="M6" height="14.4625" version="1.1" viewBox="0 0 16.75 14.4625" width="16.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.4625)"> <g transform="translate(72,-60.43)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝔹</tspan> </text> <text transform="matrix(1,0,0,-1,-63.5,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan> </text> </g> </g> </svg>.

Multiplicative Isometries on F‐Algebras of Holomorphic Functions

We study multiplicative isometries on the following F‐algebras of holomorphic functions: Smirnov class N*(X), Privalov class Np(X), Bergman‐Privalov class and Zygmund F‐algebra NlogβN(X), where X is the open unit ball 𝔹n or the open unit polydisk 𝔻n in ℂn.

Toeplitz Operators on the Dirichlet Space of 𝔹n

We study the algebraic properties of Toeplitz operators on the Dirichlet space of the unit ball 𝔹n. We characterize pluriharmonic symbol for which the corresponding Toeplitz operator is normal or isometric. We also obtain descriptions of conjugate holomorphic symbols of commuting Toeplitz operators. Finally, the commuting problem of Toeplitz operators whose symbols are of the form is studied.

Schwarz‐Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball

Let BX be the unit ball in a complex Banach space X. Assume BX is homogeneous. The generalization of the Schwarz‐Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ball Bn to BX associated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.

HYPERCYCLICITY OF WEIGHTED COMPOSITION OPERATORS ON THE UNIT BALL OF ℂN

This paper discusses the hypercyclicity of weighted composi- tion operators acting on the space of holomorphic functions on the open unit ball BN of CN. Several analytic properties of linear fractional self- maps of BN are given. According to these properties, a few necessary conditions for a weighted composition operator to be hypercyclic in the space of holomorphic functions are proved. Besides, the hypercyclicity of adjoint of weighted composition operators are studied in this paper.

ON HARMONIC BLOCH SPACES IN THE UNIT BALL OF ℂn

AbstractIn this paper, our main aim is to discuss the properties of harmonic mappings in the unit ball 𝔹n. First, we characterize the harmonic Bloch spaces and the little harmonic Bloch spaces from 𝔹n to ℂ in terms of weighted Lipschitz functions. Then we prove the existence of a Landau–Bloch constant for a class of vector-valued harmonic Bloch mappings from 𝔹n to ℂn.

Compact Differences of Composition Operators in Several Variables

When φ and ψ are linear–fractional self-maps of the unit ball BN in \({{\mathbb C}^N,N\geq 1}\), we show that the difference \({C_{\varphi}-C_{\psi}}\) cannot be non-trivially compact on either the Hardy space H2(BN) or any weighted Bergman space \({A^2_{\alpha}(B_N)}\). Our arguments emphasize geometrical properties of the inducing maps φ and ψ.

On Subordination Chains with Normalization Given by a Time-dependent Linear Operator

In this paper we are concerned with solutions, in particular with univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball Bn in \({\mathbb{C}^n}\). The main result is a generalization to higher dimensions of a well known result due to Becker. Various particular cases of this result have been recently obtained for subordination chains with normalization \({Df(0,t)=e^tI_n}\) or Df(0, t) = etA, t ≥ 0, where \({A\in L(\mathbb{C}^n,\mathbb{C}^n)}\). We also determine the form of the standard solutions to the Loewner differential equation associated with generalized spirallike mappings. In the last section we obtain the form of the solution in the presence of coefficient bounds.

The commutants of analytic Toeplitz operators for several complex variables

It is proved that if ϕ is a nonconstant bounded analytic function on the unit ball Bn and continuous on Sn in Cn ,a ndψ is a bounded measurable function on Sn such that Tϕ∗ and Tψ commute, then ψ is the boundary value of an analytic function on Bn. In addition, the commutants of two Toeplitz operators are also discussed.

On the rigidity theorem for harmonic functions in Kähler metric of Bergman type

The paper gives a method to generate the potential functions which can induce Kahler metrics u = u\( u_{i\bar j} \)dzi ⊗d\( \bar z_j \) of Bergman type on the unit ball Bn in ℂn. The paper proves that if h ∈ ℂn(\( \bar B_n \)) is harmonic in these metrics u (Δuh = 0) in Bn, then h must be pluriharmonic in Bn. In fact, it is a characterization theorem, as a consequence, the paper provides a way to construct many counter examples for the potential functions of the metric u so that the above theorem fails. The results in this paper generalize the theorems of Graham (1983) and examples constructed by Graham and Lee (1988).

A CHARACTERIZATION OF M-HARMONICITY

If f is M-harmonic and integrable with respect to a weighted radial measure fi over the unit ball Bn of C n , then R Bn (f - ˆ) dfi = f(ˆ(0)) for every ˆ 2 Aut(Bn). Equivalently f is fixed by the weighted Berezin transform; Tfif = f. In this paper, we show that if a function f defined on Bn satisfies R(f - `) 2 L 1 (Bn) for every ` 2 Aut(Bn) and Sf = rf for some |r| = 1, where S is any convex combination of the iterations of Tfi 0 s, then f is M-harmonic.

Weighted Anisotropic Integral Representations of Holomorphic Functions in the Unit Ball of Cn

We obtain weighted integral representations for spaces of functions holomorphic in the unit ball Bn and belonging to area‐integrable weighted Lp‐classes with “anisotropic” weight function of the type , w = (w1, w2, …, wn) ∈ Bn. The corresponding kernels of these representations are estimated, written in an integral form, and even written out in an explicit form (for n = 2).

Lacunary series in mixed norm spaces on the ball and the polydisk

We characterize lacunary series in mixed norm spaces on the unit ball Bn in Cn and on the unit polydisk Dn in Cn. 2010 Mathematics Subject Classifications. 32A36, 32A37. .

Homeomorphic extension of strongly spirallike mappings in ℂ n

In this paper we consider a proper subclass ŜAn of the full class of spirallike mappings on the Euclidean unit ball Bn in ℂn with respect to a given linear operator A. We use the method of subordination chains to obtain an upper growth result for ŜAn, and we obtain various examples of mappings in the same class of normalized biholomorphic mappings on the unit ball Bn in ℂn. We also prove that the class ŜAn is compact, while the full class of spirallike mappings with respect to a linear operator need not be compact in dimension n ⩾ 2, even when the operator is diagonal. This is one of the motivations for considering the class ŜAn. Finally we prove that if f is a quasiregular strongly spirallike mapping on Bn such that |[Df(z)]−1Af(z)| is uniformly bounded on Bn, then f extends to a homeomorphism of ℝ2n onto itself. In addition, if A+A* = 2aIn for some a > 0, this extension is also quasiconformal on ℝ2n.