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

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 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.