The majorization function and the index of invariant sub spaces in the Bergman spaces

The index of an invariant subspace ℳ of a Banach space of analytic functions in the open unit disc is defined by indℳ = dimℳ/zℳ. In this paper, we provide a strong link between high index and nontangential boundary behaviour of the functions in an invariant subspace. Work of the first author was partially supported by the Royal Swedish Academy of Sciences; work of the second and third authors was supported by the National Science Foundation.     

Traces of pluriharmonic functions on curves

We prove that, if γ is a simple smooth curve in the unit sphere inC ⁿ, the space o pluriharmonic functions in the unit ball, continuous up to the boundary, has a trace of finite cof dimension in the space of all continuous functions on the curve. First author partially supported by the Swedish Natural Science Research Council. Second author partially supported by CICYT grant PB85-0374.     

Subspace theorems for homogeneous polynomial forms

We prove a subspace theorem for homogeneous polynomial forms which generalizes Schmidt’s subspace theorem for linear forms. Further, we formalize the subspace theorem into a form which is just the counterpart of a second main theorem in Nevanlinna’s theorem, and also suggest a problem. The work of the first author was partially supported by NSFC of China: Project. No. 10371064. The second author was partially supported by a UGC Grant of Hong Kong: Project No. 604103.     

Lower order eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem for the Dirichlet Laplacian on a domain in an n-dimensional compact Riemannian manifold. First we give a general inequality for eigenvalues. As one of its applications, we study eigenvalues of the Laplacian on a domain in an n-dimensional complex projective space, on a compact complex submanifold in complex projective space and on the unit sphere. By making use of the orthogonalization of Gram–Schmidt (QR-factorization theorem), we construct trial functions. By means of these trial functions, estimates for lower order eigenvalues are obtained. Qing-Ming Cheng research was partially supported by a Grant-in-Aid for Scientific Research from JSPS. Hejun Sun and Hongcang Yang research were partially supported by NSF of China.     

A combination theorem for Veech subgroups of the mapping class group

In this paper we prove a combination theorem for Veech subgroups of the mapping class group analogous to the first Klein–Maskit combination theorem for Kleinian groups in which two Fuchsian subgroups are amalgamated along a parabolic subgroup. As a corollary, we construct subgroups of the mapping class group (for all genera at least 2), which are isomorphic to non-abelian closed surface groups in which all but one conjugacy class (up to powers) is pseudo-Anosov. Received: October 2004 Revision: April 2005 Accepted: April 2005 C.J.L.’s work was partially supported by an NSF postdoctoral fellowship. A.W.R’s work was partially supported by an NSF grant.     

On the optimal control for variational inequalities

We prove an existence theorem for the optimal control of variational inequalities governed by a pseudomonotone operator: the cost is assumed to be quadratic. Then, we give an extension of the theorem to more general costs (assuming the operator to be monotone); we also give a result on a perturbation problem.This work is an extended part of the author's thesis, written under the direction of Professor T. Zolezzi. This research was partially supported by the Consiglio Nazionale delle Ricerche (CNR), Rome, Italy.     

Quasiconformal Lipschitz maps, Sullivan's convex hull theorem and Brennan's conjecture

We show that proving the conjectured sharp constant in a theorem of Dennis Sullivan concerning convex sets in hyperbolic 3-space would imply the Brennan conjecture. We also prove that any conformal mapf:D→Ω can be factored as aK-quasiconformal self-map of the disk (withK independent of Ω) and a mapg:D→Ω with derivative bounded away from zero. In particular, there is always a Lipschitz homeomorphism from any simply connected Ω (with its internal path metric) to the unit disk. The author is partially supported by NSF Grant DMS 9800924.     

Q p spaces in strictly pseudoconvex domainsspaces in strictly pseudoconvex domains

We extend the definition ofQ _p spaces from the unit disk to a strictly pseudoconvex domainD in ℂ ⁿ and show that several known properties are true also in the several variable case. We also provide some proofs and examples that are new even when restricted to the one-dimensional case. Both authors partially supported by the Swedish Natural Science Research Council.     

Estimates for integral kernels of mixed type,fractional integration operators,and optimal estimates for the\bar \partial operator

Sharp tangential Lipschitz estimates for the inhomogeneous Cauchy Riemann equations with L^p data on strongly pseudoconvex doma ins in complex manifolds are proved. Estimates in both isotropic and non-isotropic spaces of functions of bounded mean oscillation are proved.Tangential estimates for a large class of domains are shown to follow from those on the ball.In the course of the proofs a fractional integration theorem of independent interest is proved.This work was partially supported by NSF Grant # MCB 77-02213     

On finite convolutions of singular distributions and a “singular analog” of the Jessen-Wintner theorem

We study the fractal properties of a convolution of two Cantor distributions. By using the method of characteristic functions, we establish sufficient conditions for the singularity of the convolution of an arbitrary finite number of distributions of random variables with independent s-adic digits. We disprove the hypothesis on the validity of a “singular analog” of the Jessen-Wintner theorem for anomalously fractal distributions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1082–1088, August, 1998. The present work was partially supported by the International Soros Program of Educational Support in Exact Sciences (grant No. APU 061086).     

Composition Operators on the Space of Bounded Harmonic Functions

We study composition operators on the space of bounded harmonic functions on the open unit disk. The principal goal of this paper is to provide criteria for determining the essential norm of difference of two composition operators. The third author is partially supported by Grant-in-Aid for Scientific Research (No.17540169), Japan Society for the Promotion of Science.     

Semigroups of Composition Operators in BMOA and the Extension of a Theorem of Sarason

In this paper we deal with the maximal subspace in BMOA where a general semigroup of analytic functions on the unit disk generates a strongly continuous semigroup of composition operators. Particular cases of this question are related to a well-known theorem of Sarason about VMOA. Our results describe analytically that maximal subspace and provide a condition which is sufficient for the maximal subspace to be exactly VMOA. A related necessary condition is also proved in the case when the semigroup has an inner Denjoy-Wolff point. As a byproduct we provide a generalization of the theorem of Sarason. This research has been partially supported by the Ministerio de Educación y Ciencia projects n. MTM2006-14449-C02-01 and MTM2005-08350-C03-03 and by La Consejería de Educación y Ciencia de la Junta de Andalucía.     

Letter to the Editor: A Short Complex-Variable Proof of the Titchmarsh Convolution Theorem

The Titchmarsh convolution theorem is a celebrated result about the support of the convolution of two functions. We present a simple proof based on the canonical factorization theorem for bounded holomorphic functions on the unit disk.

     

Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros

It is shown that new inequalities for certain classes of entire functions can be obtained by applying the Schwarz lemma and its generalizations to specially constructed Blaschke products. In particular, for entire functions of exponential type whose zeros lie in the closed lower half-plane, distortion theorems, including the two-point distortion theorem on the real axis, are proved. Similar results are established for polynomials with zeros in the closed unit disk. The classical theorems by Turan and Ankeny-Rivlin are refined. In addition, a theorem on the mutual disposition of the zeros and critical points of a polynomial is proved. Bibliography: 16 titles. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 101–112.     

Integral representation with weights I

 We describe a new approach to representation formulas for holomorphic functions, including the Cauchy-Fantappie-Leray formula, and provide a general method to generate weighted formulas. Received: 31 October 2001 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 32 A 25. The author was partially supported by the Swedish Research Council.     

Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of ℂ n

We give a Schwarz-Pick estimate for bounded holomorphic functions on unit ball in Cn, and generalize some early work of Schwarz-Pick estimates for bounded holomorphic functions on unit disk in C.     

On the sharpness of Meier’s analogue of Fatou’s theorem

Meier’s topological analogue of Fatou’s theorem is shown to be sharp by exhibiting a bounded holomorphic function in the unit disk for which no point of a prescribed set of first category on the unit circle is a Meier point. Supported by the U. S. Army Research Office, Durham.     

Families of meromorphic functions avoiding continuous functions

Leth ₁,h ₂ andh ₃ be continuous functions from the unit disk D into the Riemann sphereC such thath _i(z) ≠ h_j(z) (i ≠ j) for eachz∈D. We prove that the setF of all functionsf meromorphic on D such thatf(z)≠h _j (z) for allz ∈ D andj=1,2,3 is a normal family. The result and the method of the proof extend to quasimeromorphic functions in higher dimensions as well. The second author was supported by a Heisenberg fellowship of the DFG. The fourth author was partially supported by the Marsden Fund, New Zealand. This research was completed while the authors were attending a conference at Mathematisches Forschungsinstitut Oberwolfach in Germany. The authors would like to express their sincere thanks to the Institute for providing a stimulating atmosphere and for its kind hospitality.     

On an upper bound for the number of characteristic values of an operator function

We prove a theorem on an upper bound for the number of characteristic values of an operator-valued function that is holomorphic and bounded in a domain. This estimate is similar to the well-known inequality for zeros of a number function that is holomorphic and bounded in a domain. We derive several corollaries of the theorem proved, in particular, a statement on an estimate of the number of characteristic values of polynomial bundles of operators that lie in a given disk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 211–224, February, 1998. This work was financially supported by the Foundation for Fundamental Research of the Ukrainian Ministry of Science.     

The central limit theorem for random perturbations of rotations

Summary.   We prove a functional central limit theorem for stationary random sequences given by the transformations