Norm of the Hilbert matrix operator between some spaces of analytic functions

In this paper, we calculate the exact value of the norm of the Hilbert matrix operator $\mathcal{H}$ from the logarithmically weighted Korenblum space $H_{\alpha ,\log }^{\mathrm{\infty }}$ into Korenblum space $H_{\alpha }^{\mathrm{\infty }}$, and from the Hardy space $H^{\mathrm{\infty }}$ to the classical Bloch space $\mathcal{B}$. Furthermore, we compute the precise value of the norm on the logarithmically weighted Korenblum space $H_{\alpha ,\log }^{\mathrm{\infty }}$, and obtain both the lower and upper bounds of the norm on $\alpha$-Bloch space ${\mathcal{B}}^{\alpha }$. Finally, in the context of mapping from the Korenblum space $H_{\alpha }^{\mathrm{\infty }}$ to the $(\alpha +1)$-Bloch space ${\mathcal{B}}^{\alpha +1}$, we establish the norm of $\mathcal{H}$.