Unit Ball Of Cn Research Articles

On the Integrability of Eigenfunctions of the Laplace-Beltrami Operator in the Unit Ball of C n

Let B denote the unit ball in Cn, n≥1, and let τ, \(\widetilde\nabla\), and denote the volume measure, gradient, and Laplacian respectively, with respect to the Bergman metric on B. For γ∈R and 0<p<∞, we denote by Lpγ the set of real, or complex-valued measurable functions f on B for which ∫B(1−|z|2)γ|f(z)|p dτ(z)<∞, and by Dpγ the Dirichlet space of C1 functions f on B for which |\(\widetilde\nabla\)f|∈Lpγ. Also, for λ∈C, we denote by Xλ the set of C2 real, or complex-valued functions f on B for which f=λf. The main result of the paper is as follows: Let 0<p<∞ and suppose λ∈R with λ≥−n2. Then Lpγ∩Xλ={0}, and for λ≠0, Dpγ∩Xλ={0}

Boundary interpolation in the ball

We solve a boundary interpolation problem in the reproducing kernel Hilbert space of functions analytic in the unit ball of CN with reproducing kernel 1/(1−∑1Nzkwk*). We introduce the notion of Brune factor (or Blaschke–Potapov factor of the third kind) in this setting.

Quotients of the unit ball of ℂn for a free action of ℤ

LetBn be the unit ball of ℂn and ℤ ≅ Γ ⊂ AutBn be generated by a parabolic element of AutBn. We show that the quotientBn/Γ is biholomorphic to a holomorphically convex domain of ℂn, whose automorphism group is explicity described. It follows thatBn/ℤ is Stein for any free action of ℤ.

A New Characterization of Dirichlet Type Spaces on the Unit Ball of Cn

In this paper a BMO-type characterization of Dirichlet type spaces Dp on the unit ball of Cn is given.

Spectral lifting in Banach algebras and interpolation in several variables

Let A {\mathcal {A}} be a unital Banach algebra and let J J be a closed two-sided ideal of A {\mathcal {A}} . We prove that if any invertible element of A / J {\mathcal {A}}/J has an invertible lifting in A {\mathcal {A}} , then the quotient homomorphism Φ : A → A / J \Phi :{\mathcal {A}}\to {\mathcal {A}}/J is a spectral interpolant. This result is used to obtain a noncommutative multivariable analogue of the spectral commutant lifting theorem of Bercovici, Foiaş, and Tannenbaum. This yields spectral versions of Sarason, Nevanlinna–Pick, and Carathéodory type interpolation for F n ∞ ⊗ ¯ B ( K ) F_{n}^{\infty }\bar \otimes B({\mathcal {K}}) , the WOT-closed algebra generated by the spatial tensor product of the noncommutative analytic Toeplitz algebra F n ∞ F_{n}^{\infty } and B ( K ) B({\mathcal {K}}) , the algebra of bounded operators on a finite dimensional Hilbert space K {\mathcal {K}} . A spectral tangential commutant lifting theorem in several variables is considered and used to obtain a spectral tangential version of the Nevanlinna-Pick interpolation for F n ∞ ⊗ ¯ B ( K ) F_{n}^{\infty }\bar \otimes B({\mathcal {K}}) . In particular, we obtain interpolation theorems for matrix-valued bounded analytic functions on the open unit ball of C n \mathbb {C}^{n} , in which one bounds the spectral radius of the interpolant and not the norm.

Noncommutative interpolation and Poisson transforms

General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF ∞ (resp., noncommutative disc algebraA n) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂn are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebraF ∞/J on Hilbert spaces whereJ is anyw *-closed, 2-sided ideal ofF ∞, are obtained and used to construct aw *-continuous,F ∞/J-functional calculus associated to row contractionsT=[T 1,…,T n] whenf(T1, …, Tn)=0 for anyf∈J. Other properties of the dual algebraF ∞/J are considered.

Regularity of jacobians

It is shown that if F is a holomorphic mapping of the unit ball in Cn into Cn which has a certain smoothness (in the Sobolev sense) then the jacobian of F has somewhat more smoothness than might be expected. Examples are given to show that the results are sharp.

The Volume of the Unit Ball in Cn

The Volume of the Unit Ball in Cn

Interpolating sequences in the ball of Cn

LetB be the unit ball of Cn, I give necessary conditions on sequence S of points in B to be H∞(B) interpolating in term of a Cn valued holomorphic function zero on S (a substitute for the interpolating Blaschke product). These conditions are sufficient to prove that the sequence S is interpolating for ∩p>1 (B) and is also interpolating for Hp(B) for 1≤p<∞.

Weighted Bergman Spaces and Carleson Measures on the Unit Ball of Cn

Let φ be a normal function on [0, 1), Bn the unit ball of Cn, and Aφp(Bn) the weighted Bergman spaces on Bn with weight φ. The purpose of this paper is to discuss some relations among Aφp(Bn), weighted Bergman kernels, and Carleson measures on Bn.

A note on weighted Bergman spaces and the Cesaro Operator

Let B denote the unit ball in ℂn, and dV(z) normalized Lebesgue measure on B. For α &gt; -1, define dVα(z) = (1 - \z\2)αdV(z). Let (B) denote the space of holomorhic functions on B, and for 0 &lt; p &lt; ∞, let p(dVα) denote Lp(dVα) ∩ (B). In this note we characterize p(dVα) as those functions in (B) whose images under the action of a certain set of differential operators lie in Lp(dVα). This is valid for 1 &lt; p &lt; oo. We also show that the Cesàro operator is bounded on p(dVα) for 0 &lt; p &lt; oo. Analogous results are given for the polydisc.

Sampling sequences for Hardy spaces of the ball

We show that a sequence a := {ak}k in the unit ball of Cn is sampling for the Hardy spaces Hp, 0 1 (in contrast to the one dimensional situation) there exist sampling sequences for H∞ whose admissible accumulation set has measure 0. We also consider the sequence a(ω) obtained by applying to each ak a random rotation, and give a necessary and sufficient condition on {|ak |}k so that, with probability one, a(ω) is of sampling for Hp, p < ∞. §

RANDOM DIRICHLET TYPE FUNCTIONS ON THE UNIT BALL OF Cn

The random Dirichlet type functions on the unit ball of Cn are studied. Sufficient conditions of the multipliers of for 0 1 are given. The smoothness of random Dirichlet type functions is discussed.

On Analytic Functions of Bergman BMO in the Ball

AbstractLet B = Bn be the open unit ball of Cn with volume measure v, U = B1 and B be the Bloch space on , 1 ≤ α &lt; 1, is defined as the set of holomorphic f : B → C for whichif 0 &lt; α &lt; 1 and , the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic f : B → U for which the composition operator defined by , is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric.

Convexity properties of holomorphic mappings in ℂⁿ

Not many convex mappings on the unit ball in C n {\mathbb C}^n for n &gt; 1 n&gt;1 are known. We introduce two families of mappings, which we believe are actually identical, that both contain the convex mappings. These families which we have named the “Quasi-Convex Mappings, Types A and B” seem to be natural generalizations of the convex mappings in the plane. It is much easier to check whether a function is in one of these classes than to check for convexity. We show that the upper and lower bounds on the growth rate of such mappings is the same as for the convex mappings.

The growth theorem of convex mappings on the unit ball in ℂⁿ

Let ‖ ⋅ ‖ \Vert \cdot \Vert be an arbitrary norm on C n {\mathbb {C}}^{n} . Let f f be a normalized biholomorphic convex mapping on the unit ball in C n {\mathbb {C}}^{n} with respect to the norm ‖ ⋅ ‖ \Vert \cdot \Vert . We will give an upper bound of the growth of f f .

A Refinement of The Fatou–Naim–Doob Theorem

We introduce the notion of harmonic thin sets and establish a refinement of the Fatou–Naim–Doob Theorem in the axiomatic system of Brelot, with an added assumptiom which is fufilled for the classical Laplace operator in ℝn. We verify this assumption for the Poisson–Szego integrals on the unit ball in ℂn as well as the Weinstein equation on a halfspace in ℝn. Thus, a stronger version of the Fatou–Naim–Doob Theorem is given in those cases. The assumption is expected to be verified for a large class of second order elliptic differential operators. The application of our result to sets of determination (for harmonic functions), introduced by Beurling [2], Dahlberg [9], Bonsall [3], and Hayman and Lyons [16], will follow in a separate paper.

Common fixed points of commuting holomorphic maps in the unit ball of ℂⁿ

Let B n \mathbb {B}^n be the unit ball of C n \mathbb {C}^n ( n &gt; 1 n&gt;1 ). We prove that if f , g ∈ H o l ( B n , B n ) f,g \in \mathrm {Hol}(\mathbb {B}^n, \mathbb {B}^n) are holomorphic self-maps of B n \mathbb {B}^n such that f ∘ g = g ∘ f f \circ g = g \circ f , then f f and g g have a common fixed point (possibly at the boundary, in the sense of K K -limits). Furthermore, if f f and g g have no fixed points in B n \mathbb {B}^n , then they have the same Wolff point, unless the restrictions of f f and g g to the one-dimensional complex affine subset of B n \mathbb {B}^n determined by the Wolff points of f f and g g are commuting hyperbolic automorphisms of that subset.

Compact Toeplitz operators on Bergman spaces

Let Bn denote the open unit ball in Cn. We write V to denote Lebesgue volume measure on Bn normalized so that V(Bn)=1. Fix −1&lt;γ&lt;∞ and let Vγ denote the measure given by dVγ(z)=cγ (1−[mid ]z[mid ]2)γdV(z), for z∈Bn, where cγ=Γ(n+γ+1)/ (n!Γ(γ+1)); then Vγ(Bn)=1. The weighted Bergman space A2,γ(Bn) is the space of all analytic functions in L2(Bn, dVγ). This is a closed linear subspace of L2(Bn, dVγ). Let Pγ denote the orthogonal projection of L2(Bn, dVγ) onto A2,γ(Bn). For a function f∈L∞(Bn) the Toeplitz operator Tf is defined on A2,γ(Bn) by Tfh=Pγ(fh), for h∈A2,γ(Bn). It is clear that Tf is bounded on A2,γ(Bn) with ∥Tf∥[les ]∥f∥∞. In this paper we will consider the question for which f∈L∞(Bn) the operator Tf is compact on A2,γ(Bn). Although a complete answer has been given by the author and D. Zheng (see the next section), the condition for compactness is somewhat unnatural. In this article we will give a more natural description for compactness of Toeplitz operators with sufficiently nice symbols. We will describe compactness in terms of behaviour of the so-called Berezin transform of the symbol, which has been useful in characterizing compactness of Toeplitz operators with positive symbols (see [5, 9]). Before we can define this Berezin transform we need to introduce more notation.

A–∞-interpolation in the ball

We give a necessary and sufficient condition for a sequence {ak}k in the unit ball of ℂn to be interpolating for the class A–∞ of holomorphic functions with polynomial growth. The condition, which goes along the lines of the ones given by Berenstein and Li for some weighted spaces of entire functions and by Amar for H∞ functions in the ball, is given in terms of the derivatives of m ≥ n functions F1, …,Fm ∈ A–∞ vanishing on {ak}k.