<abstract xmlns="http://www.w3.org/1999/xhtml">

For a Bloch function <italic>f</italic> in the unit ball in ℂ<italic>n</italic>, we study the maximal locus of the Bloch norm of <italic>f</italic>; namely, the set <italic>Lf</italic> where the Bergman length of the gradient vector field of <italic>f</italic> attains its maximum. We prove that for <italic>n</italic> ≥<italic>,</italic> the set <italic>Lf</italic> consists of a finite union of real analytic sets with dimensions at most 2<italic>n −</italic> 2. This is not the case for <italic>n</italic> = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set <italic>Lf</italic>. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.
</abstract>

For a Bloch function f in the unit ball in ℂn, we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at most 2n − 2. This is not the case for n = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set Lf. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.

The Maximum Locus of the Bloch Norm

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"The Maximum Locus of the Bloch Norm"}

For a Bloch function f in the unit ball in ℂn, we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient...

The Maximum Locus of the Bloch Norm

Publié en ligne: 07 juin 2023

Pages: 291 - 303

Reçu: 22 nov. 2022

Accepté: 29 mars 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0019

Mots clés
Bergman Kernel, Bergman metric, Bloch function, Bloch

© 2023 Youssfi El Hassan, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The Maximum Locus of the Bloch Norm

Publié en ligne: 07 juin 2023

Pages: 291 - 303

Reçu: 22 nov. 2022

Accepté: 29 mars 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0019

Mots clésBergman Kernel, Bergman metric, Bloch function, Bloch

© 2023 Youssfi El Hassan, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Mots clés
Bergman Kernel, Bergman metric, Bloch function, Bloch