Norm of the Hilbert matrix operator on logarithmically weighted Bloch and Hardy spaces

In this paper, we compute the exact value of the norm of the Hilbert matrix operator $\mathcal{H}$ acting from the classical Bloch space $\mathcal{B}$ into the logarithmically weighted Bloch space ${\mathcal{B}}_{\log }$, and show that it equals $\frac{3}{2}$; we also find that the norm from the space of bounded analytic functions $H^{\mathrm{\infty }}$ into the logarithmically weighted Hardy space $H_{\log }^{}$ is $1$. Furthermore, we establish both lower and upper bounds for the norm of $\mathcal{H}$ when it maps from the $\alpha$-Bloch space ${\mathcal{B}}^{\alpha }$ into the logarithmically weighted ${\mathcal{B}}_{\log }^{}$ with $1<\alpha <2$, and from the Hardy space $H^1$ into the logarithmically weighted Hardy space $H_{\log }^{}$.