The exceptional sets for functions of the Bergman space in the unit ball

Piotr Jakóbczak

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993)

  • Volume: 4, Issue: 2, page 79-85
  • ISSN: 1120-6330

Abstract

top
Let D be a domain in C 2 . Given w C , set D w = z C z , w D . If f is a holomorphic and square-integrable function in D , then the set E D , f of all w such that f ( , w ) is not square-integrable in D w has measure zero. We call this set the exceptional set for f . In this Note we prove that whenever 0 < r < 1 there exists a holomorphic square-integrable function f in the unit ball B in C 2 such that E B , f is the circle C 0 , r = z C z = r .

How to cite

top

Jakóbczak, Piotr. "The exceptional sets for functions of the Bergman space in the unit ball." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 4.2 (1993): 79-85. <http://eudml.org/doc/244307>.

@article{Jakóbczak1993,
abstract = {Let \( D \) be a domain in \( C^\{2\} \). Given \( w \in C \), set \( D\_\{w\} = \\{ z \in C \mid (z,w) \in D\\} \). If \( f \) is a holomorphic and square-integrable function in \( D \), then the set \( E(D,f) \) of all \( w \) such that \( f ( \cdot, w) \) is not square-integrable in \( D\_\{w\} \) has measure zero. We call this set the exceptional set for \( f \). In this Note we prove that whenever \( 0 < r < 1 \) there exists a holomorphic square-integrable function \( f \) in the unit ball \( B \) in \( C^\{2\} \) such that \( E(B,f) \) is the circle \( C(0, r) = \\{z \in C \mid |z| = r\\} \).},
author = {Jakóbczak, Piotr},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Bergman space; Hartogs domain; Exceptional sets; Bergman space in the unit ball; exceptional sets},
language = {eng},
month = {6},
number = {2},
pages = {79-85},
publisher = {Accademia Nazionale dei Lincei},
title = {The exceptional sets for functions of the Bergman space in the unit ball},
url = {http://eudml.org/doc/244307},
volume = {4},
year = {1993},
}

TY - JOUR
AU - Jakóbczak, Piotr
TI - The exceptional sets for functions of the Bergman space in the unit ball
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1993/6//
PB - Accademia Nazionale dei Lincei
VL - 4
IS - 2
SP - 79
EP - 85
AB - Let \( D \) be a domain in \( C^{2} \). Given \( w \in C \), set \( D_{w} = \{ z \in C \mid (z,w) \in D\} \). If \( f \) is a holomorphic and square-integrable function in \( D \), then the set \( E(D,f) \) of all \( w \) such that \( f ( \cdot, w) \) is not square-integrable in \( D_{w} \) has measure zero. We call this set the exceptional set for \( f \). In this Note we prove that whenever \( 0 < r < 1 \) there exists a holomorphic square-integrable function \( f \) in the unit ball \( B \) in \( C^{2} \) such that \( E(B,f) \) is the circle \( C(0, r) = \{z \in C \mid |z| = r\} \).
LA - eng
KW - Bergman space; Hartogs domain; Exceptional sets; Bergman space in the unit ball; exceptional sets
UR - http://eudml.org/doc/244307
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.