Integral logarithmic means of Blaschke products

Using the Fourier series method, we generalize results characterizing the behavior of the integral logarithmic means of Blaschke products of arbitrary order for meromorphic functions.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 199–206, August, 1998.     

On the boundedness of the total variation of the logarithm of a Blaschke product

We establish that, for a Blaschke product B(z) convergent in the unit disk, the condition - ∞ < smallint ₀¹ log(1 - t)n(t,B)dtsmallint _0^1 log (1 - t)n(t,B)dt is sufficient for the total variation of logB to be bounded on a circle of radiusr, 0 <r < 1. For products B(z) with zeros concentrated on a single ray, this condition is also necessary. Here, n(t, B) denotes the number of zeros of the functionB (z) in a disk of radiust.     

Weak infinite products of Blaschke products

We study weak infinite products for sequences of Blaschke products. Using properties of these functions, -subsets of zero sets of functions in are studied. An affirmative answer is given to a problem on prime ideals of posed by Gorkin and Mortini.

     

Centralizers of finite Blaschke products

In this article we consider the set of finite Blaschke productsF of degreen>1. We give sufficient conditions forF to commute only with their own powers among all rational functions of the Riemann sphere, in terms of the derivate around the fixed points.The author was partially supported by CNPq, Brazil.     

Cluster sets of interpolating Blaschke products

We construct interpolating Balschke products whose radial cluster sets at a given point of the unit circle can be prescribed to be one of the following: the closed unit disk; an arbitrary closed arc on the unit circle; an arbitrary interval of the form [x, y], wherexy ≠ 0 and −1≤1x≤y≤1. We also show that there does not exist an interpolating Blaschke product having [0,y] or [x, 0] as a radial cluster set. On the other hand, there do exist finite products of interpolating Blaschke products that have [0, 1] as a radial cluster set. Research supported by the RIP-program Oberwolfach, 2002/2003.     

Weak infinite powers of Blaschke products

Letb be a Blaschke product with zeros {z _n } in the open unit disk Δ. Let be the set of sequences of non-negative integersp=(p ₁,p ₂,…) such that ∑ _n=1 ^∞ p _n (1 − |z _n |) < ∞ andp _n →∞ asn→∞. We study the class of weak infinite powers ofb, Properties of these classes depend on the setS(b) of the cluster points in ∂Δ of {z _n }. It is proved thatS(b)=∂Δ if and only if , the Douglas algebra generated by . Also, it is proved thatdθ(S(b))=0 if and only if there exists an interpolating Blaschke productB such that .     

Radial limits of interpolating Blaschke products

It is shown that if (_n) is a sequence of distinct points on the unit circle, then, for a sequence (a_n) of points in the closed unit disk, there exists an interpolating Blaschke product B with B*(_n)=a_n for all n if and only if (a_n) is bounded away from zero. This complements results of Cargo, Carroll, Colwell, Belna and Piranian on prescribing radial limits for Blaschke products.Mathematics Subject Classification (2000): 30D50Revised version: 1 June 2004Acknowledgment The authors thank the Mathematisches Forschungsinstitut Oberwolfach for the support and for the kind hospitality they always receive. The work presented here is part of their 2002–2004 project Inner functions. The first author also thanks Universität Bern where she spent her sabbatical, as well as Université de Metz, where she was professeur invité for one month. The second author presented this work in May 2004 at the fourth congress of the European network Analysis and Operators in Dalfsen, the Netherlands. He gratefully acknowledges the support he received from the European communitys human potential program, contract HPRN-CT-2000-00116.     

On the Blaschke conjecture for 3-webs

We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This resolves the Blaschke conjecture for 3-webs. We also give the algorithm for determining whether a given 3-web is linearizable, find the linearity condition for 3-webs and establish its relationships to the condition that a plane curve consists of flexes and to the Euler equation in gas-dynamics.     

Blaschke products and expanding maps of the circle

For an analytic map of the closed unit disk onto itself that leaves the boundary circle invariant, we give necessary and sufficient conditions for the map of the boundary to be expanding.

     

On Blaschke products with derivatives in Bergman spaces with normal weights

We generalize a well-known sufficient condition for interpolating sequences for the Hilbert Bergman spaces to other Bergman spaces with normal weights (as defined by Shields and Williams) and obtain new results regarding the membership of the derivative of a Blaschke product or a general inner function in such spaces. We also apply duality techniques to obtain further results of this type and obtain new results about interpolating Blaschke products.     

Two new characterizations of carleson-newman Blaschke products

We present an extension of a result of Vasyunin by giving a characterization of finite products of interpolating Blaschke products B in terms of the minorization of B(z) by the distance of z to the zeros of B. We also characterize those Blaschke products that satisfy the hereditary weak embedding property.     

Absolutely continuous conjugacies of Blaschke products III

Research supported in part by the National Science Foundation.     

Boundary spectra of uniform Frostman Blaschke products

A closed set in the unit circle is the boundary spectrum of a uniform Frostman Blaschke product if and only if is nowhere dense in .

     

On the boundedness of the maximal operators of double Walsh-logarithmic means of Marcinkiewicz type

The main aim of this paper is to investigate the (H _p , L _p )-type inequality for the maximal operators of Riesz and Nörlund logarithmic means of the quadratical partial sums of Walsh-Fourier series. Moreover, we show that the behavior of Nörlund logarithmic means is worse than the behavior of Riesz logarithmic means in our special sense.     

Quasi-symmetric conjugacy of Blaschke products on the unit circle

Let B be a Blaschke product for which the restriction to theunit circle ¹ is a degree d > 1 covering. We prove that Bis quasi-symmetrically conjugate to z z^d on ¹, if all its periodicpoints in ¹ are repelling and if ¹ does not intersect the -limitset of any recurrent critical point for B.     

On the boundedness of bilinear operators on products of Besov and Lebesgue spaces

We prove mapping properties of the form and , for certain related indices p₁,p₂,p₃,q₁,q₂,α₁,α₂∈R, where T is a bilinear Hörmander-Mihlin multiplier or a molecular paraproduct. Applications to bilinear Littlewood-Paley theory are discussed.