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

Abstract: 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 $H^\infty$ into the logarithmically weighted Hardy space $H^{\infty}{\log}$ 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 $\mathcal{B}^\alpha_{\log}$ with $1 <\alpha < 2$, and from the Hardy space $H^{1}$ into the logarithmically weighted Hardy space $H^{1}_{\log}$.