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Single point extremal functions in Bergman-type spaces

Aleman, Alexandru LU and Richter, S (2002) In Indiana University Mathematics Journal 51(3). p.581-605
Abstract
Let A be a zero sequence for the Bergman space L-a(2) of the unit disc D, and let phi(A) be the corresponding canoniacal zero divisor. In this paper we consider quotients of the type phi(Au {alpha})/phi(A), alpha is an element of D. By use of methods from the theory of reproducing kernels we shall show that the modulus of such functions is always bounded by 3, and that they can be written as a product of a single Blaschke factor and a function whose real part is greater than 1. Our methods apply in somewhat larger generality. In particular, our results lead to a new proof of the contractive zero-divisor property in weighted Bergman spaces with logarithmically subharmonic weights. For the unweighted Bergman spaces L-a(p), 0 < p <... (More)
Let A be a zero sequence for the Bergman space L-a(2) of the unit disc D, and let phi(A) be the corresponding canoniacal zero divisor. In this paper we consider quotients of the type phi(Au {alpha})/phi(A), alpha is an element of D. By use of methods from the theory of reproducing kernels we shall show that the modulus of such functions is always bounded by 3, and that they can be written as a product of a single Blaschke factor and a function whose real part is greater than 1. Our methods apply in somewhat larger generality. In particular, our results lead to a new proof of the contractive zero-divisor property in weighted Bergman spaces with logarithmically subharmonic weights. For the unweighted Bergman spaces L-a(p), 0 < p < infinity, we show that the canonical zero divisor phi(A) for a zero sequence with n elements can be written as a product of n starlike functions. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Bergman spaces, contractive divisors
in
Indiana University Mathematics Journal
volume
51
issue
3
pages
581 - 605
publisher
Indiana University
external identifiers
  • wos:000176918700003
  • scopus:0036059736
ISSN
1943-5258
DOI
10.1512/iumj.2002.51.2190
language
English
LU publication?
yes
id
b40a0935-9ab9-4935-99a5-ea4ce359480c (old id 333305)
date added to LUP
2007-08-06 14:30:12
date last changed
2017-01-01 05:13:41
@article{b40a0935-9ab9-4935-99a5-ea4ce359480c,
  abstract     = {Let A be a zero sequence for the Bergman space L-a(2) of the unit disc D, and let phi(A) be the corresponding canoniacal zero divisor. In this paper we consider quotients of the type phi(Au {alpha})/phi(A), alpha is an element of D. By use of methods from the theory of reproducing kernels we shall show that the modulus of such functions is always bounded by 3, and that they can be written as a product of a single Blaschke factor and a function whose real part is greater than 1. Our methods apply in somewhat larger generality. In particular, our results lead to a new proof of the contractive zero-divisor property in weighted Bergman spaces with logarithmically subharmonic weights. For the unweighted Bergman spaces L-a(p), 0 &lt; p &lt; infinity, we show that the canonical zero divisor phi(A) for a zero sequence with n elements can be written as a product of n starlike functions.},
  author       = {Aleman, Alexandru and Richter, S},
  issn         = {1943-5258},
  keyword      = {Bergman spaces,contractive divisors},
  language     = {eng},
  number       = {3},
  pages        = {581--605},
  publisher    = {Indiana University},
  series       = {Indiana University Mathematics Journal},
  title        = {Single point extremal functions in Bergman-type spaces},
  url          = {http://dx.doi.org/10.1512/iumj.2002.51.2190},
  volume       = {51},
  year         = {2002},
}