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

Authors

  • Mikael Lindström Åbo Akademi University, Department of Mathematics
  • Santeri Miihkinen Åbo Akademi University, Department of Mathematics
  • Niklas Wikman Åbo Akademi University, Department of Mathematics

Abstract

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 Karapetrovic. We obtain a complete solution to the conjecture for \(\alpha > 0\) 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.
Section
Articles

Published

2021-06-24

How to Cite

Lindström, M., Miihkinen, S., & Wikman, N. (2021). On the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces. Annales Fennici Mathematici, 46(1), 201–224. Retrieved from https://afm.journal.fi/article/view/109770