Distribution of martingales with bounded square functions

We study the terminate distribution of a martingale whose square function is bounded. We obtain sharp estimates for the exponential and p-moments, as well as for the distribution function itself. The proofs are based on the elaboration of the Burkholder method and on the investigation of certain locally concave functions.     

The universal central extension of the holomorphic current algebra

We identify the universal differential module ¹(A) for the Fréchet algebra A of holomorphic functions on a complex Stein manifold X, and more generally on a Riemannian domain R over X and for the algebra of germs of holomorphic functions on a compact subset K ⁿ . It turns out to be isomorphic to the Fréchet space of holomorphic 1-forms on X, resp. R, resp. to the space ¹(K) of germs of holomorphic 1-forms in K. This determines the center of the universal central extension of the Lie algebra (R, of holomorphic maps from R to a finite-dimensional simple complex Lie algebra .     

Onn-widths of bounded periodic holomorphic functions

The even-dimensional Kolmogorov widthsd _2n , Gel'fand widthsd ²ⁿ , and linear widths _2n ofà inL _q andC are determined exactly. We show that all threen-widths are equal and give a characterization of the widths in terms of Blaschke products.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1170–1175, September, 1995.     

The spectrum of an algebra of weakly continuous holomorphic mappings

Let U be an arbitrary absolutely convex open subset of a complex Banach space E and let F be a Banach algebra with identity. The spectrum of the algebra H_b(U, F) of the holomorphic mappings from U and F which are bounded on the U-bonded subsets of U is studied in case E′ has the approximation properly.     

Conjugate operators that preserve the nonconvergence of bounded martingales

We show that the conjugate T^∗ of an operator , with X and Y Banach spaces, satisfies the following dichotomy: either T^∗ preserves the nonconvergence of bounded martingales in Y^∗, or there exists a compact operator such that the kernel N(T^∗+K^∗) fails the Radon-Nikodým property.     

Algebras generated by two bounded holomorphic functions

We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, one of which is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension (of the closure) of such algebras. The conditions are expressed in terms of the inner part of a certain function which is explicitly derived from each pair of generators. Our results are based on identifyingz-invariant subspaces included in the closure of the algebra. The second-named author thanks the University at Albany, Harvard University, and Brown University for their hospitality during the completion of this work.     

On bounded holomorphic interpolation in several variables

We show how the reproducing kernel Hilbert space approach to some holomorphic interpolation problems can be utilized to proveboundedness of their solutions. The goal is two-fold: i) to propose a several variable analogue of the classical Pick-Nevanlinna theorem, ii) to display that a simple argumentation may be carried on in this case, which, though still in a Hilbert space context, avoids any use of operator theory.     

Hilbert spaces of holomorphic functions on bounded domains

In [6] the notion of Hilbert-Stein complex spaces was introduced. The purpose of this note is to illustrate the conjecture that any Stein space whose points can be separated by bounded holomorphic functions, is Hilbert-Stein. We prove this fact for the following kinds of bounded domains of holomorphyD in C ⁿ : 1)n=1 and ₁(D) is finitely generated, 2) the group of automorphisms ofD in the compactopen topology is compact, 3)D is strongly pseudoconvex with smooth boundary andn2, 4)D is symmetric of type I.     

Universality of holomorphic functions bounded on closed sets

In this note, the existence of translation-universal entire functions which are bounded on certain closed subsets is characterized in terms of topological and geometrical properties of such subsets. Corresponding results are also stated in the space of holomorphic functions on the unit disk and in the space of harmonic functions on the plane. Moreover, it is shown the existence of entire functions which are bounded on many rays and, simultaneously, are universal with respect to a prescribed infinite-order differential operator.     

Spectra of algebras of holomorphic functions of bounded type

We prove that if U is a balanced H_b (U)-domain of holomorphy in Tsirelson's space then the spectrum of H_b (U) is identified with U. We derive theorems of Banach Stone type for algebras of holomorphic functions and algebras of holomorphic germs.     

Functional calculus for generators of uniformly bounded holomorphic semigroups

We present a method for constructing a functional calculus for (possibly unbounded) operators that generate a uniformly bounded holomorphic semigroup, e^−zA. (A will be called a generator.) These are closed, densely defined operators whose spectrum and numerical range are contained in [0,∞), with respect to an equivalent norm.     

Games of bounded deviation,additivizations of games and asymptotic martingales

We consider games in coalition function form on a, generally infinite, algebra of coalitions. For finite algebras the additive part mappingv E(v ¦) is the usual. The concern here is the analogue for infinite algebras. The useful construction is the finitely additive stochastic process of additive parts of the game on the filtration _f of finite subalgebras of.It is shown that is an isomorphism between: