Zero sets of holomorphic functions in the unit ball with slow growth

We study the zero-varieties of holomorphic functions in the unit ball satisfying the growth condition log |f(z)|≤c _fλ(|z|), where λ:(0,1)→ℝ⁺ is a positive increasing function. We obtain some sufficient conditions on an analytic variety to be defined by such a function. Some results for the particular case λ(r)=log(e/(1−r)), corresponding to the classA ^−∞, generalize those of B. Korenblum in one variable. Both authors supported by DGICYT grant PB92-0804-C02-02.     

Equiaffine geometry of level sets and ruled hypersurfaces with equiaffine mean curvature zero

Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of 2n‐dimensional nondegenerate hypersurfaces ruled by n‐planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with respect to which the ruling is Lagrangian.     

Polar sets on metric spaces

We show that if is a proper metric measure space equipped with a doubling measure supporting a Poincaré inequality, then subsets of with zero -capacity are precisely the -polar sets; that is, a relatively compact subset of a domain in is of zero -capacity if and only if there exists a -superharmonic function whose set of singularities contains the given set. In addition, we prove that if is a -hyperbolic metric space, then the -superharmonic function can be required to be -superharmonic on the entire space . We also study the the following question: If a set is of zero -capacity, does there exist a -superharmonic function whose set of singularities is precisely the given set?

     

Spherical limits for Riesz potentials of functions in central generalized Orlicz-Morrey spaces on the unit ball

We introduce central generalized Orlicz–Morrey spaces on the unit ball, and study the weighted behavior of spherical means for Riesz potentials of functions in those spaces. We also treat Orlicz–Morrey–Sobolev functions which are monotone in the punctured unit ball in the sense of Lebesgue.     

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.     

Zero sets of holomorphic functions in the bidisk

We characterize in geometric terms the zero sets of holomorphic functionsf in the bidisk such that log |f|∈L ^p (D ²) for 1<p<∞. Partially supported by the DGCYT grant PB95-0956-C02-02 and grant 1996-SGR-26.     

Extreme harmonic functions on a ball

In order to show that one can recapture the Riesz-Herglotz theorem from the Krein-Milman theorem, we determine directly the set of extreme points of the convex set of positive harmonic functions on the unit ball (normalized by 1 at the origin). The characterization is obtained using standard facts from abstract analysis combined with a minimum of very basic results on harmonic functions.     

On the central limit theorem for the stationary Poisson process of compact sets

Stochastic geometry models based on a stationary Poisson point process of compact subsets of the Euclidean space are examined. Random measures on ?^d, derived from these processes using Hausdorff and projection measures are studied. The central limit theorem is formulated in a way which enables comparison of the various estimators of the intensity of the produced random measures. Approximate confidence intervals for the intensity are constructed. Their use is demonstrated in an example of length intensity estimation for the segment processes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Functional properties of privalov spaces of holomorphic functions in several variables

We consider Privalov classes of degreeq>1 in the unit ball and the polydisk in ℂⁿ. They are defined, say, for the ball, as the sets of functionsf(z) such that the average of ln ₊ ^q |f(z)| over a sphere centered at the origin remains bounded as the radius increases to 1. These classes, which were introduced (in the one-dimensional case) by Privalov before 1941, were often used in the foreign literature in the last 10–20 years; typically, the notation varied and Privalov was not mentioned. We discuss various equivalent definitions of these classes as well as the most general properties, such as growth estimates, properties of the natural metric, and boundedness or total boundedness of subsets. Translated fromMatematicheskie Zametki Vol. 65, No. 2, pp. 280–288, February, 1999.     

On a theorem of Privalov and normal functions

A well known result of Privalov asserts that if is a function which is analytic in the unit disc , then has a continuous extension to the closed unit disc and its boundary function is absolutely continuous if and only if belongs to the Hardy space . In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, we prove that for any positive continuous function defined in with , as , there exists a function analytic in which is not a normal function and with the property that , for all sufficiently close to .

     

Intersection of sets with -connected unions

We show that if sets in a topological space are given so that all the sets are closed or all are open, and for each every of the sets have a -connected union, then the sets have a point in common. As a consequence, we obtain the following starshaped version of Helly's theorem: If every or fewer members of a finite family of closed sets in have a starshaped union, then all the members of the family have a point in common. The proof relies on a topological KKM-type intersection theorem.

     

Hyperbolic mean growth of bounded holomorphic functions in the ball

We consider the hyperbolic Hardy class , . It consists of holomorphic in the unit complex ball for which and

where denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type -function and the area function are defined in terms of the invariant gradient of , and membership of is expressed by the property of the functions. As an application, we can characterize the boundedness and the compactness of the composition operator , defined by , from the Bloch space into the Hardy space .

     

On zero varieties of holomorphic functions in Hardy spaces

In contrast to the famous Henkin-Skoda theorem concerning the zero varieties of holomorphic functions in the Nevanlinna class on the open unit ball B_n in , n?2, it is proved in this article that for any nonnegative, increasing, convex function ?(t) defined on , there exists satisfying such that there is no f∈H^p(B_n), 0<p<∞, with . Here N_g(ζ,1) denotes the integrated zero counting function associated with the slice function g_ζ. This means that the zero sets of holomorphic functions belonging to the Hardy spaces H^p(B_n), 0<p<∞, unlike that of the holomorphic functions in the Nevanlinna class, cannot be characterized in the above manner.     

Zeroes of slice monogenic functions

In this article, we study the structure of zeroes of power series with Clifford algebra‐valued coefficients. Especially, if it has paravector‐valued coefficients, we obtain some sufficient and necessary conditions of power series that have zeroes, as well as a method to compute the zeroes if exist. Copyright 2011 John Wiley & Sons, Ltd.     

A family of permitted trigonometric thin sets

We introduce the notion of perfectly measure zero sets and prove that every perfectly measure zero set is permitted for the families of all pseudo-Dirichlet sets, N-sets, A-sets and N-sets. In particular this means that these families of trigonometric thin sets are closed under adding sets of cardinality less than the additivity of Lebesgue measure.

     

On the zero sets of certain entire functions

We consider the class of entire functions of the form

where are polynomials and are entire functions. We prove that the zero-set of such an , if infinite, cannot be contained in a ray. But for every region containing the positive ray there is an example of with infinite zero-set which is contained in this region.

     

The limit functions of a random iteration system

This paper discusses the limit functions of a random iteration system formed by finitely many rational functions. Applying these results we prove that a hyperbolic iteration system has no wandering domain and that its limit functions are constant. Finally the continuity on its Julia set is considered.