<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

Published Online: Jun 07, 2023

Page range: 291 - 303

Received: Nov 22, 2022

Accepted: Mar 29, 2023

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

Keywords
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

Published Online: Jun 07, 2023

Page range: 291 - 303

Received: Nov 22, 2022

Accepted: Mar 29, 2023

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

KeywordsBergman 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.

Keywords
Bergman Kernel, Bergman metric, Bloch function, Bloch