A criterion of normality based on a single holomorphic function

Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f ^(k)(z)| ≤ c, where c is a constant. Then F is normal on D.     

Restricted T-universal functions on multiply connected domains

Let G ⊂ C be a domain which is bounded by a finite number of pairwise disjoint Jordan curves. We prove the existence of a function which is holomorphic exactly on G and has universal translates with respect to a prescribed set E ⊂∂G and which in addition is continuous on G ^-\E. This revised version was published online in June 2006 with corrections to the Cover Date.     

Tensor products of Drinfeld modules andv-adic representations

We define the tensor product ϕ ⊗ ψ and relatedt-modules Sym²(ϕ), and ∧²(ϕ) for Drinfeld modules ϕ, ψ defined over the rational function fieldK=F _q (T), and describe thev-adic Tate modules of theset-modules by using those of ϕ, ψ.     

Realization of a sum of sequences by a sum graph

It is shown that the realizability of the sequences ϕ=(a ₁,…, _a ), ψ=(b ₁,…,b _n ) and ϕ+ψ is a sufficient condition for the realizability of ϕ+ψ by a graph with a ϕ-factor ifb _i ≦1 fori=1,…,n. The condition is not sufficient in general. A necessary and sufficient condition for the realizability of ϕ+ψ by a graph with a ϕ-factor is given for the case that ϕ is realizable by a star and isolated vertices.     

Essential norm of the difference of composition operators on Bloch space

Let ϕ and ψ be holomorphic self-maps of the unit disk, and denote by C _ϕ , C _ψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators C _ϕ −C _ψ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.     

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993,Tsal proved 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,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.     

Boundary functions on a bounded balanced domain

We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.      

Holomorphic functions which are highly nonintegrable at the boundary

Given a bounded convex domainD in ℂ^N with smooth boundary and a positive continuous function ϕ ond, it is proved that there is a holomorphic functionf onD such that |f|ϕ is nonintegrable onM∩D wheneverM is a real submanifold of a neighbourhood of a point ofbD which intersectsbD transversely.     

A holomorphic extension theorem from a fractal hypersurface in ℂ<Superscript>2</Superscript>

We consider the problem of identifying boundary values of holomorphic functions on bounded domains in ℂ². We use the quaternionic analysis techniques to extending the CR structure to a pure function theoretical nature. The advantage of our procedure lies in the fact that it also runs for domains with fractal boundary.     

The bounded approximation property for spaces of holomorphic mappings on infinite dimensional spaces

We study the bounded approximation property for spaces of holomorphic functions. We show that if U is a balanced open subset of a Fréchet–Schwartz space or (DFM )‐space E , then the space ??(U ) of holomorphic mappings on U , with the compact‐open topology, has the bounded approximation property if and only if E has the bounded approximation property. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

cos πρ theorems for δ-subharmonic functions

For ϕ a δ-subharmonic function, sharp results are obtained that connectA(r, ϕ), B(r, ϕ) andT(r, ϕ), whereA(r, ϕ)=inf_|z|=r ϕ(z),B(r, ϕ)=sup_|z|=r ϕ(z), andT(r, ϕ) is the Nevanlinna characteristics.     

Boundary functions in L2 H $$(\mathbb{B}^n )$$

We solve the Dirichlet problem for line integrals of holomorphic functions in the unit ball For a function u which is lower semi-continuous on we give necessary and sufficient conditions in order that there exists a holomorphic function f ∈ such that

Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point

For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H ₀ : f≡ 0, (the signal f is zero) against the nonparametric alternative H ₁ : f∈Λ_ɛ where Λ_ɛ is a set of functions on R ¹ of the form Λ_ɛ = {f : f∈?, ϕ(f) ≥ Cψ_ɛ}. Here ? is a H?lder or Sobolev class of functions, ϕ(f) is either the sup-norm of f or the value of f at a fixed point, C > 0 is a constant, ψ_ɛ is the minimax rate of testing and ɛ→ 0 is the asymptotic parameter of the model. We find exact separation constants C ^* > 0 such that a test with the given summarized asymptotic errors of first and second type is possible for C > C ^* and is not possible for C < C ^*. We propose asymptotically minimax test statistics. Received: 23 February 1998 / Revised version: 6 April 1999 / Published online: 30 March 2000     

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 question of Gol’dberg concerning entire functions with prescribed zeros

Let (z_j) be a sequence of complex numbers satisfying |z_j|→ ∞ asj → ∞ and denote by n(r) the number of z_j satisfying |z_j|≤ r. Suppose that lim inf_{r → ⇈} log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫^∞ (ϕ(t)t logt)⁻¹ dt < ∞. It is proved that there exists an entire functionf whose zeros are the z_j such that log log M(r,f) = o((log n(r))²ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.     

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f ^(k)(z)| < |h(z)|; (b) f ^(k)(z) ≠ h(z). Then F is normal on D.     

Immagini inverse dei gruppi di omotopia

Riassunto SeM edN sono varietà poliedriche chiuse connesse ed orientate di dimensioni rispettivem edn, conm≥n>2, edf∶M→N è una trasformazione continua, allora per ognir, minore din e non inferiore a 2, si definisce un omomorfismo indotto ϕ_rπ:_r (N)→H _m-n+r (M) dal quale si ricavano certi invarianti topologici.

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Résumé Soientm≥n>r≥2 des entiers etM, N des variétés polyédrales closes connexes orientées satisfaisant dimM=m et dimN=n, de plusH _i(M) le groupe de Betti à i dimensions deM,M,π _i (N) le groupe de Hurewicz ài dimensions deN, etf∶M→N une application continue. Alorsf définit, pour,r=2, 3, …n−1, un homomorphisme réciproque ϕ_rπ:_r (N)→H _m-n+r (M) comme il suit. Etant donné un élément α du groupe π_r (N) et uner-sphère continue orientéeS de α, on peut supposer quef ⁻¹(S) soit un polyèdre finiA àm−n+r dimensions. Parf est induit dansA un (m−n+r)-cyclez à coefficients entiers, et la classe d'homologie dez est justement l'image ϕr(α) de α par ϕr. Pourr=1, on obtient un homomorphisme réciproque ϕ_rπ:_r (N)→H _m-n+r (M) du groupe fondamentalF(N) deN dans le groupe d'homologie àm−n+1 dimensions deM. A l'aide des homomorphismes ϕ,,ϕ2,ϕ,3...,ϕn-i, on parvient à certaines expressions caractéristiques dépendantes seulement de la classe d'homotopie def, en particulier on obtient des constantes pour les images des bases de Betti deM, pour Fimage du groupe de torsion deM, et pour l'image réciproque du groupe fondamental deN.

     

Interpolation of operators of weak type (ϕ<Subscript>0</Subscript>, ψ<Subscript>0</Subscript>, ϕ<Subscript>1</Subscript>, ψ<Subscript>1</Subscript>) in Lorentz spaces

We prove theorems on interpolation of quasilinear operators of weak type (ϕ₀, ψ₀, ϕ₀, ψ₁) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and convex functions ϕ₀(t), ψ₀(t), ϕ₁(t), and ψ₁(t). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005.     

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP _n (z, f, f′)=P _n−1(z, f, f′, ... f ^(p)), whereP _n andP _n−1 are polynomials in all variables, the degree ofP _n with respect tof andf′ is equal ton, and the degree ofP _n−1 with respect tof, f′, ... f ^(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E _*, whereE _* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z ^1/2 (β+o(1)), β∈C, holds forz=re ^iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re ^iν)‖=o(r ^1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.     

The sum of coefficients of bounded univalent functions

We solve the maximal value problem for the functional in the class of functionsf(z)=z+a ₂z²+… that are holomorphic and univalent in the unit disk and satisfy the inequality |f(z)|<M. We prove that the Pick functions are extremal for this problem for sufficiently largeM whenever the set of indicesk ₁,…,k_m contains an even number. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 728–733, May, 1997. Translated by S. S. Anisov