Existence of Universal Taylor Series for Nonsimply Connected Domains

It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that possess “universal” Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in ℂ\Ω that have connected complement. This paper shows, for nonsimply connected domains Ω, how issues of capacity, thinness and topology affect the existence of holomorphic functions on Ω that have universal Taylor series expansions about a given point.     

Indices of Toeplitz tuples on pseudoregular domains

This paper proves an index theorem of Toeplitz tuples on pseudoregular domains in Cn. Geometrically, the index of Toeplitz tuple TΦn is (-1)n time wrapping number of Φn around the origin. As one of the applications of the index theorem, we completely characterize the automorphism groups of Toeplitz algebras on Poincaré domain. As another application, it is shown that C*(Ω)C*(Bn) for every Poincare domain Ω in Cn(n≠2). It is also noticed that C*(Ω)C*(B2) if and only if the Poincaré conjecture is true for Ω.     

The Gleason’s problem for some polyharmonic and hyperbolic harmonic function spaces

Let Ω ⊆ ℝⁿ be a bounded convex domain with C ² boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space H _k ^p,q,φ (Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥_p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, H _k ^p,q,φ ) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.     

Composition Operators on the Bloch Space of Several Complex Variables

Abstract In this paper, we study the boundedness and compactness of composition operator C _φ on the Bloch space β(Ω), Ω being a bounded homogeneous domain. For Ω = B _n, we give the necessary and sufficient conditions for a composition operator C _φ to be compact on β(B _n) or β ₀(B _n). Supported by the National Natural Science Foundation and the National Education Committee Doctoral Foundation     

On the Asymptotic Behavior of Faber Polynomials for Domains With Piecewise Analytic Boundary

Let φ(z) be an analytic function on a punctured neighborhood of ∞, where it has a simple pole. The nth Faber polynomial F _n (z) (n=0,1,2,…) associated with φ is the polynomial part of the Laurent expansion at ∞ of [φ(z)] ⁿ . Assuming that ψ (the inverse of φ) conformally maps |w|>1 onto a domain Ω bounded by a piecewise analytic curve without cusps pointing out of Ω, and under an additional assumption concerning the “Lehman expansion” of ψ about those points of |w|=1 mapped onto corners of ∂ Ω, we obtain asymptotic formulas for F _n that yield fine results on the limiting distribution of the zeros of Faber polynomials.      

Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle

Motivated by problems arising from Arithmetic Geometry, in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric. In the case of a germ of holomorphic isometry f: (Δ, λ ds _Δ²;0) → (Ω, ds _Ω²;0) of the Poincaré disk Δ into a bounded symmetric domain Ω ⋐ ℂ ^N in its Harish-Chandra realization and equipped with the Bergman metric, f extends to a proper holomorphic isometric embedding F: (Δ, λ ds _Δ²;) → (Ω, ds _Ω²) and Graph(f) extends to an affine-algebraic variety V ⊂ ℂ × ℂ ^N . Examples of F which are not totally geodesic have been constructed. They arise primarily from the p-th root map ρ _p : H → H ^p and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H ₃ of genus 3. In the current article on the one hand we examine second fundamental forms σ of these known examples, by computing explicitly φ = |σ|². On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincaré disk into polydisks. For such mappings expressing via the inverse Cayley transform in terms of the Euclidean coordinate τ = s + it on the upper half-plane H, we have φ(τ) = t ² u(τ), where u| _t=0 ≢ 0. We show that u must satisfy the first order differential equation δu/δt| _t=0 ≡ 0 on the real axis outside a finite number of points at which u is singular. As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincaré disk into the polydisk must develop singularities along the boundary circle. The equation δu/δt| _t=0 ≡ 0 along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from G. Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article.     

Weighted composition operators on growth spaces

Denote by Hol(B _n ) the space of all holomorphic functions in the unit ball B _n of ℂ ⁿ , n ≥ 1. Given g ∈ Hol(B _m ) and a holomorphic mapping φ: B _m → B _n , put C _φ ^g f = g · (f ∘ φ) for f ∈ Hol(B _n ). We characterize those g and φ for which C _φ ^g is a bounded (or compact) operator from the growth space A ^−log(B _n ) or A ^−β (B _n ), β > 0, to the weighted Bergman space A _α ^p (B _m ), 0 < p < ∞, α > −1. We obtain some generalizations of these results and study related integral operators.     

Pseudo-dimension and entropy of manifolds formed by affine-invariant dictionary

We consider the manifolds H _n(φ) formed by all possible linear combinations of n functions from the set {φ(A⋅+b)}, where x→Ax+b is arbitrary affine mapping in the space ℝ^d. For example, neural networks and radial basis functions are the manifolds of type H _n(φ). We obtain estimates for pseudo-dimension of the manifold H _n(φ) for wide collection of the generator function φ. The estimates have the order O(d ² n) in degree scale, that is the order is proportional to number of parameters of the manifold H _n(φ). Moreover the estimates for ɛ-entropy of the manifold H _n(φ) are obtained. Mathematics subject classifications (2000) 41A46, 41A50, 42A61, 42C10 V. Maiorov: Supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

Bounded universal functions for sequences of holomorphic self-maps of the disk

We give several characterizations of those sequences of holomorphic self-maps {φ _n } _n≥1 of the unit disk for which there exists a function F in the unit ball of H ^∞ such that the orbit {F∘φ _n :n∈ℕ} is locally uniformly dense in . Such a function F is said to be a -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ _n . As a consequence we will see that if φ _n is the nth iterate of a map φ of into , then {φ _n } _n≥1 admits a -universal function if and only if φ is a parabolic or hyperbolic automorphism of . We show that whenever there exists a -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.     

D p spaces on bounded symmetric domains ofC n

In this paper, the spaceD ^p(Ω) of functions holomorphic on bounded symmetric domain ofC ⁿ is defined. We prove thatH ^p(Ω)⊂D ^p(Ω) if 0<p≤2, andD ^p(Ω)⊂H ^p(Ω) ifp≥2, and both the inclusions are proper. Further, we find that some theorems onH ^p(Ω) can be extended to a wider classD ^p(Ω) for 0<p≤2.     

On the number of real critical points of logarithmic derivatives and the Hawaii conjecture

For a given real entire function φ in the class U _2n ^*, n ≥ 0, with finitely many nonreal zeroes, we establish a connection between the number of real zeroes of the functions Q[φ] = (φ′/φ)′ and Q ₁[φ] = (φ″/φ′)′. This connection leads to a proof of the Hawaii Conjecture (T. Craven, G. Csordas, and W. Smith [5]), which states that if φ is a real polynomial, then the number of real zeroes of Q[φ] does not exceed the number of nonreal zeroes of φ.     

Compact composition operators on the Bloch space in polydiscs

Let Uⁿ be the unit polydisc of ℂⁿ and φ=(φ₁, ⋯, φ _n ) a holomorphic self-map of Uⁿ. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Uⁿ) if and only if for every ε > 0, there exists a δ > 0, such that

Generalization of theorems of Szász and Ruscheweyh on exact bounds for derivatives of analytic functions

Let Ω and Π be two domains in the extended complex plane equipped by the Poincaré metric. In this paper we obtain analogs of Schwarz-Pick type inequalities in the class A(Ω, gH) = {f: Ω → Π} of functions locally holomorphic in Ω; for the domain Ω we consider the exterior of the unit disk and the upper half-plane. The obtained results generalize the well-known theorems of Szász and Ruscheweyh about the exact estimates of derivatives of analytic functions defined on the disk |z| < 1.     

A direct proof for lower semicontinuity of polyconvex functionals

Lower semicontinuity for polyconvex functionals of the form ∫_Ω g(detDu)dx with respect to sequences of functions fromW ^1,n (Ω;ℝ ⁿ ) which converge inL ¹ (Ωℝ ⁿ ) and are uniformly bounded inW ^1,n−1 (Ω;ℝ ⁿ ), is proved. This was first established in [5] using results from [1] on Cartesian Currents. We give a simple direct proof which does not involve currents. We also show how the method extends to prove natural, essentially optimal, generalizations of these results. Supported by MURST, Gruppo Nazionale 40% Partially supported by Australian Research Council     

Nonexistence of Proper Holomorphic Maps Between Certain Classical Bounded Symmetric Domains

The author,motivated by his results on Hermitian metric rigidity,conjectured in [4] that a proper holomorphic mapping f：Ω→Ω′from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ′is necessarily totally geodesic provided that r′：=rank（Ω′）≤rank（Ω）：=r.The Conjecture was resolved in the affirmative by I.-H.Tsai [8].When the hypothesis r′≤r is removed,the structure of proper holomorphic maps f：Ω→Ω′is far from being understood,and the complexity in studying such maps depends very much on the difference r′-r,which is called the rank defect.The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H.Tu [10],in which a rigidity theorem was proven for certain pairs of classical domains of type I,which implies nonexistence theorems for other pairs of such domains.For both results the rank defect is equal to 1,and a generaliza- tion of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of （Ω→Ω′） of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and L-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case.     

A structure theorem for boundary-transitive graphs with infinitely many ends

We prove a structure theorem for locally finite connected graphsX with infinitely many ends admitting a non-compact group of automorphisms which is transitive in its action on the space of ends, Ω _X . For such a graphX, there is a uniquely determined biregular treeT (with both valencies finite), a continuous representationφ : Aut(X) → Aut(T) with compact kernel, an equivariant homeomorphism λ : Ω _X → Ω _T , and an equivariant map τ : Vert(X) → Vert(T) with finite fibers. Boundary-transitive trees are described, and some methods of constructing boundary-transitive graphs are discussed, as well as some examples.     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999     

Bounded projectors in spaces of functions holomorphic in the unit ball

The paper studies some bounded operators in the Banach spaces L ^∞(B) and L ¹(B) over the unit ball B of ℂ ⁿ , the range of which are the corresponding holomorphic subspaces A _∞(φ) and A ¹(ϕ) depending on a normal pair of weight-functions {φ, ϕ}.     

Hartogs-Bochner type theorem in projective space

We prove the following Hartogs-Bochner type theorem: Let M be a connected C² hypersurface of P_n(C) (n≥2) which divides P_n(C) in two connected open sets Ω₁ and Ω₂. Suppose that M has at most one open CR orbit. Then there exists i∈{1,2} such that C¹ CR functions defined on M extends holomorphically to Ω _i . Supported by the TMR network.