On the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces

• Published in: arXiv.org 2020-01
• Main Authors: Lindström, Mikael, Miihkinen, Santeri, Wikman, Niklas
• Format: Article
• Language: English
• ISSN: 2331-8422

In this article, the open problem of finding the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces \(A^p_\alpha\) is adressed. The norm was conjectured to be \(\frac{\pi}{\sin \frac{(2+\alpha)\pi}{p}}\) by Karapetrovi\'{c}. We obtain a complete solution to the conjecture for \(\alpha \ge 0\) and \(2+\alpha+\sqrt{\alpha^2+\frac{7}{2}\alpha+3} \le p < 2(2+\alpha)\) and a partial solution for \(2+2\alpha < p < 2+\alpha+\sqrt{\alpha^2+\frac{7}{2}\alpha+3}.\) Moreover, we also show that the conjecture is valid for small values of \(\alpha\) when \(2+2\alpha < p \le 3+2\alpha.\) Finally, the case \(\alpha = 1\) is considered.