Unit Ball Of Cn Research Articles

Generalized Forelli-Rudin type operators between several function spaces on the unit ball of ℂn

Generalized Forelli-Rudin type operators between several function spaces on the unit ball of ℂn

Essential norm of Riemann–Stieltjes operator on weighted Bergman spaces with doubling weights

Abstract Let ω be a doubling weight and 0 < p ≤ q < ∞ {0<p\leq q<\infty} . The essential norm of Riemann–Stieltjes operator T g {T_{g}} from the weighted Bergman space A ω p {A^{p}_{\omega}} to A ω q {A^{q}_{\omega}} was investigated in the unit ball of ℂ n {\mathbb{C}^{n}} .

The Gleason’s problem on normal weight general function spaces in the unit ball of ℂn

The Gleason’s problem on normal weight general function spaces in the unit ball of ℂn

Spectral theorem approach to commutative C⁎-algebras generated by Toeplitz operators on the unit ball: Quasi-elliptic related cases

We consider commutative C⁎-algebras of Toeplitz operators in the weighted Bergman space on the unit ball in Cn. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.

The Maximum Locus of the Bloch Norm

Abstract For a Bloch function f in the unit ball in ℂ n , we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at most 2n − 2. This is not the case for n = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set Lf . In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.

Area operators on Hardy spaces in the unit ball of [formula omitted

We characterize boundedness and compactness of area operators from Hp into Lq(Sn) in terms of Carleson measures. Some of the tools used in the proof of the one dimensional case are not available in the higher dimension case, such as the strong factorization of Hardy spaces and the Calderón-Zygmund decomposition. Our approach involves a duality argument and maximal functions of Lp(Sn) functions in the unit ball of Cn.

Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions

The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in Rn into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in Cn or from the unit polydisc in Cn into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficient type Schwarz-Pick inequalities for pluriharmonic functions defined in the Minkowski space. Finally, we use the obtained Schwarz-Pick type inequalities to discuss the Lipschitz continuity, the Schwarz-Pick type lemmas of arbitrary order and the Bohr phenomenon of harmonic or pluriharmonic functions.

Toeplitz operators from hardy spaces to weighted Bergman spaces in the unit ball of Cn

Toeplitz operators from hardy spaces to weighted Bergman spaces in the unit ball of Cn

Geometric Arveson-Douglas conjecture for the Hardy space and a related compactness criterion

We consider a class of analytic subsets M˜ of an open neighborhood of the closed unit ball in Cn. Such an M˜ gives rise to a submodule R and a quotient module Q of the Hardy module H2(S) on the unit sphere S⊂Cn. We show that, as predicted by the geometric Arveson-Douglas conjecture, the quotient module Q is p-essentially normal for p>d=dimCM˜. We further show that, more interestingly, the quotient module Q exhibits a behavior that is only found on the Bergman space and the Fock space: an operator A in the Toeplitz algebra on Q is compact if and only if its Berezin transform vanishes near M˜∩S.

$${\cal N}\left( {p,q,s} \right)$$-Type Spaces in the Unit Ball of ℂn. IV: Atomic Decomposition, Gleason’s Problem and Distance Problems

The purpose of this paper is to study several different properties of the holomorphic $${\cal N}\left( {p,q,s} \right)$$ -type functions in the unit ball $$\mathbb{B}$$ of ℂn via Carleson measure techniques. More precisely, we establish an atomic decomposition on such spaces and then we solve the Gleason’s problem on them. We also study the distance problems between Bergman-type spaces $${A^{ - {q \over p}}}\left(\mathbb{R} \right)$$ and $${\cal N}\left( {p,q,s} \right)$$ -type spaces. These results yield some new characterizations of the holomorphic F(p, q, t)-functions.

Complex symmetric monomial Toeplitz operators on the unit ball

Monomial Toeplitz operators Tzpz¯q on the weighted Bergman space of the unit ball in Cn are natural extension of the weighted shift operators. In higher dimensions, dim ker⁡Tzpz¯q=∞ may occur for non-trivial case, thus complex symmetric phenomena will appear. In this paper, we completely characterize when the monomial Toeplitz operator Tzpz¯q on the weighted Bergman space is JA-symmetric, where JAf(z)=f(Az‾)‾ with the symmetric unitary n×n matrix A. Moreover, we also determine all possible forms of A such that Tzpz¯q is JA-symmetric. As an application, we construct the weakly closed unital commutative algebra generated by complex symmetric Toeplitz operators.

Difference of Weighted Composition Operators on Weighted-Type Spaces in the Unit Ball

In this paper, a new characterization is provided for the boundedness, compactness and essential norm of the difference of two weighted composition operators on weighted-type spaces in the unit ball of ℂn.

Loewner chains and nonlinear resolvents of the Carathéodory family on the unit ball in [formula omitted

In this paper we study various properties of nonlinear resolvents of holomorphic mappings in the Carathéodory family M(Bn), where Bn is the Euclidean unit ball in Cn. First, we prove certain characterizations of inverse Loewner chains f(z,t)=e−∫0ta(τ)dτz+⋯ on Bn×[0,∞), where a:[0,∞)→C is a locally Lebesgue integrable function such that ℜa(t)>0 for a.e. t≥0, and which satisfies a natural assumption. Next, we prove that if f∈M(Bn), then the nonlinear resolvent family {Jr}r≥0 is an inverse Loewner chain on Bn and the associated Herglotz vector field is of divergence type, where Jr=Jr[f]=(In+rf)−1, r≥0. This result is a generalization to higher dimensions of a recent result due to Elin, Shoikhet and Sugawa. We prove that (1+r)Jr can be embedded as the first element of a normal Loewner chain and the family {(1+r)Jr[f]:0≤r<∞,f∈M(Bn)} is a compact subset of S0(Bn). Also, we deduce that the shearing of (1+r)Jr (r≥0) associated with the family M(B2) is quasi-convex of type A and also starlike of order 4/5. We give a sufficient condition for (1+r)Jr to be quasiconformal on Bn and to be extended to a quasiconformal homeomorphism of Cn onto itself. Finally, sharp coefficient bounds for the nonlinear resolvent families of certain subsets of M(Bn), and also examples of support points of the compact families generated by nonlinear resolvent mappings on B2, will be obtained.

Doubling chains on complements of algebraic hypersurfaces

A doubling chart on an n-dimensional complex manifold Y is a univalent analytic mapping ψ: B1 → Y of the unit ball in ℂn, which is extendable to the (say) four times larger concentric ball of B1. A doubling covering of a compact set G in Y is its covering with images of doubling charts on Y. A doubling chain is a series of doubling charts with non-empty subsequent intersections. Doubling coverings (and doubling chains) provide, essentially, a conformally invariant version of Whitney’s ball coverings of a domain W ⊂ ℝn, introduced in [19] (compare [11]). We study doubling chains in the complement Y = ℂnH of a complex algebraic hypersurface H of degree d in ℂn, and provide information on their length and other properties. Our main result is that any two points v1, v2 in a distance δ from H can be joined via a doubling chain in the complement Y = ℂnH of length at most c1$$\log \left( {\frac{{{c_2}}}{\delta }} \right)$$ with explicit constants c1, c2 depending only on n and d. As a consequence, we obtain an upper bound on the Kobayashi distance in Y, and an upper bound for the constant in a doubling inequality for regular algebraic functions on Y. We also provide the corresponding lower bounds for the length of the doubling chains, through the doubling constant of specific functions on Y.

Angular derivatives and boundary values of H(b) spaces of unit ball of ℂn

In this work, deBranges–Rovnyak spaces, , on the unit ball of are studied. An integral representation of the functions in through the Clark measure on associated with b is given and a characterization of admissible boundary limits is given in relation with finite angular derivatives. Lastly, the interplay between Clark measures and angular derivatives is examined and it is obtained that Clark measure associated with b has an atom at a boundary point if and only if b has finite angular derivative at the same point.

Pointwise Multiplication Operators from Hardy Spaces to Weighted Bergman Spaces in the Unit Ball of ℂn

This article is devoted to characterizing the boundedness and compactness of multiplication operators from Hardy spaces to weighted Bergman spaces in the unit ball of ℂn.

Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic Lie algebra on finite dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. Our first main result is the construction of an explicit differential operator intertwining the bundle with the direct sum of its factors. Next, we study Hilbert spaces of sections of these bundles. We use this to get, in particular, a full description and a similarity theorem for homogeneous n-tuples of operators in the Cowen-Douglas class of the Euclidean unit ball in Cn.

Dynamics of weighted composition operators in the unit ball

In the present paper, we investigate the dynamic behavior of weighted composition operators acting on the space of holomorphic functions on the unit ball in CN.

An integral estimate and the equivalent norms on F(p, q, s, k) spaces in the unit ball

In this article, the authors give a typical integral's bidirectional estimates for all cases. At the same time, several equivalent characterizations on the F(p, q, s, k) space in the unit ball of Cn are given.

Herglotz functions of several quaternionic variables

We first review realizations of Herglotz functions in the unit ball of CN and provide new insights. Then, we define the corresponding class and prove the extend the results in the case of several quaternionic variables.