BiBloch-type Maps: Existence and Beyond

It is shown that for f > 0 there are two constants c 1 , c 2 > 0 and a holomorphic map f from the unit disk ${\shadD} {\rm of} {\shadC}$ into ${\shadC}^2$ such that $ c_1(1-|z|)^{-\alpha }\le |\,f'(z)|\le c_2(1-|z|)^{-\alpha } $ for all $ z\in {\shadD}$ . Moreover, this existence is effectively used in the study of invariance of the Bloch-type spaces under composition, but also in the discussion of embedding the Bloch-type spaces via derivation into the Lebesgue, mixed-norm and Coifman-Meyer-Stein tent spaces.     

Biholomorphically invariant families amongst Carleson class

Given α (0, 1), let T_α be the Carleson class of all meromorphic maps ƒ from the unit disk to the extended complex plane with

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where ƒ^# and dm mean the spherical derivative of ƒ and Lebesgue area measure on separately. And, let BIT_α and BIT_α,0 be the biholomorphically invariant families (amongst the Carleson class) consisting of those ƒ T_α with sup and lim_{|w| → 1} ||ƒ ο φ_w||_Tα = 0 respectively, where

. The main purpose of this article is to study BIT_α and BIT_α,0 via the Ahlfors-Shimizu characteristic, canonical factorization and bounded holomorphic maps.     

The range of holomorphic maps at boundary points

We prove a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains. That is, we prove that the local image of a holomorphic map \(f{:\,}D\rightarrow D'\) close to a boundary regular contact point \(p\in \partial D\) where the Jacobian is bounded away from zero along normal non-tangential directions has to eventually contain every cone (and more generally every region which is Kobayashi asymptotic to a cone) with vertex at \(f(p)\) .     

Sets of Approximation and Interpolation in ℂ for Manifold-Valued Maps

We give examples of non-smooth sets in the complex plane with the property that every holomorphic map continuous to the boundary from these sets into any complex manifold may be uniformly approximated by maps holomorphic in some neighborhood of the set (Mergelyan-type approximation for manifold-valued maps.) Similar results are proved for sections of complex-valued holomorphic submersions from complex manifolds.      

(1/α)-Self similar α-stable processes with stationary increments

In this note we settle a question posed by Kasahara, Maejima, and Vervaat. We show that the α-stable Lévy motion is the only (1/α)-self-similar α-stable process with stationary increments if 0 < α < 1. We also introduce new classes of (1/α)-self-similar α-stable processes with stationary increments for 1 < α < 2.     

A simple proof of a theorem by Uhlenbeck and Yau

A subbundle of a Hermitian holomorphic vector bundle (E, h) can be metrically and differentially defined by the orthogonal projection onto itself. A weakly holomorphic subbundle of (E, h) is, by definition, an orthogonal projection π lying in the Sobolev space L²₁ of L² sections of End E with L² first order derivatives in the sense of distributions, which satisfies furthermore (Id−π)∘D′′π=0. A weakly holomorphic subbundle of (E, h) is shown to define a coherent subsheaf of (E), and implicitly a holomorphic subbundle of E in the complement of an analytic subset of codimension ≥2. This result provided the key technical argument to the proof given by Uhlenbeck and Yau for the Kobayashi-Hitchin correspondence on compact Kähler manifolds. We give here a much simpler proof based on current theory. The idea is to construct local meromorphic sections of Im π which locally span the fibers. We first make this construction on one-dimensional submanifolds of X and subsequently extend it by means of a Hartogs-type theorem of Shiffman’s.     

Cohomological relation between Jacobi forms and skew‐holomorphic Jacobi forms

Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito‐Kurokawa conjecture. Later Skoruppa introduced skew‐holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between Jacobi forms and skew‐holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on and the space of skew‐holomorphic Jacobi cusp forms on with the same half‐integral weight to the Eichler cohomology group of with a coefficient module coming from polynomials.     

Inner functions in Cn

In this paper we prove that if f is a holomorphic function in a strictly pseudoconvex region D in Cⁿ, n > 1, with radial limit equal to 1 in modulus at each point of some nonempty open subset S of the boundary of D, then f const in D.Translated from Matematicheskii Zametki, Vol. 19, No. 1, pp. 63–66, January, 1976.I wish to thank B. V. Shabat for a discussion of the result and for critical remarks.     

On regularly branched maps

Let be a perfect map between finite-dimensional metrizable spaces and p1. It is shown that the space of all bounded maps from X into with the source limitation topology contains a dense G_δ-subset consisting of f-regularly branched maps. Here, a map is f-regularly branched if, for every n1, the dimension of the set is n(dimf+dimY)−(n−1)(p+dimY). This is a parametric version of the Hurewicz theorem on regularly branched maps.     

Multiplicative Maps With Specified (1, 1)-Entry

In this article, we will show that for many complex-valued maps f over a semigroup G of matrices, there exists a minimum k for the existence of a multiplicative map for which the (1, 1)-entry of ϕ( A ) is f ( A ). We obtain results on such multiplicative maps, and use them to classify all the multiplicative maps τon G such that f ( A ) = τ( g ( A )) for any A ε G , where f and g are given complex-valued maps.     

Weakly Cyclic Vectors with a Given Modulus

Let H^p be the Hardy space in the polydisk. Denote by the set of all holomorphic polynomials. A vector f ∈ H^p is called weakly cyclic if the product f is weakly dense in H^p, 0 < p < 1. We construct weakly cyclic vectors with a prescribed lower semi-continuous modulus of the boundary values. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 111–118.     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

On functions with one‐dimensional holomorphic extension property in circular domains

In this paper we consider continuous functions given on the boundary of a circular bounded domain D in , , and having the one‐dimensional holomorphic extension property along family of complex lines, passing through a finite number of points of D. We study the problem of existence of holomorphic extension of such functions into D.     

Proper maps which are Lipschitz α up to the boundary

In this paper we shall construct proper holomorphic mappings from strictly pseudoconvex domains in Cⁿ into the unit ball in C^N which satisfy some regularity conditions up to the boundary. If we only require continuity of the map, but not more, then there is a large class of such maps (see [2], [3], and [5]). On the other hand, if F is C^k on the closure, k > N ? n + 1, then there is a very small class of such maps. In fact such F must be holomorphic across the boundary (see [1] and [4]). We are interested in maps F that are less than C^N ? n + 1, but more than continuous on the closure. Namely, we want to find out if this is a very small or a large class. Our main result is as follows. Theorem, (a) Let ga < 1/6; then there exists an N = N(α, n) such that we can find a map F: Bⁿ → B^N that is proper, holomorphic, and Lipschitz α up to the boundary, but F is not holomorphic across the boundary. (b) If D is a general strictly pseudoconvex domain with C^∞ -boundary in Cⁿ, then we can find a map F: D → B^N, N = N(α, n), that is proper, holomorphic, and Lipschitz α up to the boundary of D. To do part (a) of the theorem we only need to show that we can find a proper holomorphic map F = (f₁, …, F_N): Bⁿ → B^N that is Lipschitz α and f_N(z) = c(1 - Z₁)^1/6 for some constant c > 0. With this we can in fact ensure that the map in (a) is at most Lipschitz 1/6 on the closure of Bⁿ.     

Asplund operators and holomorphic maps

LetE andF be locally convex topological vector spaces. A holomorphic mapf: E→F is defined to be an Asplund map if it takes the separable subsets of a neighbourhood of eacha∈E into absolutely convex weakly metrisable subsets ofF; a Banach space is an Asplund space if and only if its identity map has this property. We show that a continuous linear map from a quasinormable locally convex spaceE into a Banach spaceF is an Asplund map if and only if it factors through an Asplund space. IfE andF are both Banach spaces, then a holomorphic mapf: E→F is an Asplund map if and only if its derivative maps factor through Asplund spaces for eacha∈E. This is true if and only if such a factorisation holds ata=0. Part of this research was done during a visit to the University of Namibia, whose financial support is gratefully acknowledged This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990     

On vector bundles of finite order

We define Finsler metrics of finite order on a holomorphic vector bundle by imposing estimates on the holomorphic bisectional curvature. We generalize the vanishing theorem of Griffiths and Cornalba regarding Hermitian bundles of finite order to the Finsler context. We develop a value distribution theory for holomorphic maps from the projectivization of a vector bundle to projective space. We show that the projectivization of a Finsler bundle of finite order can be immersed into a projective space of sufficiently large dimension via a map of finite order.     

A sharp Schwarz inequality on the boundary

A number of classical results reflect the fact that if a holomorphic function maps the unit disk into itself, taking the origin into the origin, and if some boundary point maps to the boundary, then the map is a magnification at . We prove a sharp quantitative version of this result which also sharpens a classical result of Loewner.

     

The Faber Operator and its Boundedness

Let G be a domain bounded by a Jordan curve Γ, and let A(G) be the Banach space of functions continuous on G and holomorphic in G. The Faber operator T is a linear mapping from A( ) to A(G) mapping wⁿ onto the nth Faber polynomial F_n(z) (n=0, 1, 2, …). We show that T<∞ if Γ is piecewise Dini-smooth, and give an example of a quasicircle Γ for which T=∞.     

Subconvexity bounds for Rankin–Selberg L-functions for congruence subgroups

Estimation of shifted sums of Fourier coefficients of cusp forms plays crucial roles in analytic number theory. Its known region of holomorphy and bounds, however, depend on bounds toward the general Ramanujan conjecture. In this article, we extended such a shifted sum meromorphically to a larger half plane Res>1/2 and proved a better bound. As an application, we then proved a subconvexity bound for Rankin–Selberg L-functions which does not rely on bounds toward the Ramanujan conjecture: Let f be either a holomorphic cusp form of weight k, or a Maass cusp form with Laplace eigenvalue 1/4+k², for . Let g be a fixed holomorphic or Maass cusp form. What we obtained is the following bound for the L-function L(s,fg) in the k aspect:

L(1/2+it,fg)k^{1−1/(8+4θ)+ε},