Norm estimates of weighted composition operators pertaining to the Hilbert Matrix

Very recently, Bo\v{z}in and Karapetrovi'c solved a conjecture by proving that the norm of the Hilbert matrix operator $\mathcal{H}$ on the Bergman space $A^p$ is equal to $\frac{\pi }{\sin (\frac{2\pi }{p})}$ for $2<p<4.$ In this article we present a partly new and simplified proof of this result. Moreover, we calculate the exact value of the norm of $\mathcal{H}$ defined on the Korenblum spaces $H_{\alpha }^{\mathrm{\infty }}$ for $0<\alpha \le 2\mathrm{/}3$ and an upper bound for the norm on the scale $2\mathrm{/}3<\alpha <1$.