Norm of the Hilbert matrix operator on logarithmically weighted Bloch and Hardy spaces
Shanli Ye, Qisong Zheng
TL;DR
The paper determines the exact operator norms of the Hilbert matrix operator $\mathcal{H}$ on several spaces of analytic functions with logarithmic weighting. By representing $\mathcal{H}$ as an average of weighted composition operators $T_t$, the authors derive precise norm estimates and, where possible, exact values: $\|\mathcal{H}\|_{\mathcal{B}\to\mathcal{B}_{\log}}=\tfrac{3}{2}$ and $\|\mathcal{H}\|_{H^{\infty}\to H^{\infty}_{\log}}=1$, along with lower and upper bounds for $\|\mathcal{H}\|_{\mathcal{B}^{\alpha}\to\mathcal{B}^{\alpha}_{\log}}$ for $1<\alpha<2$ and for $\|\mathcal{H}\|_{H^{1}\to H^{1}_{\log}}$. The work also establishes boundedness in the $H^1$ to $H^1_{\log}$ setting with sharp-ish bounds $\pi\le\|\mathcal{H}\|_{H^{1}\to H^{1}_{\log}}\le 2\pi$, and demonstrates non-boundedness outside certain parameter ranges for the $\alpha$-Bloch spaces. Overall, the results furnish exact and sharp operator-norm information for $\mathcal{H}$ on a spectrum of logarithmically weighted analytic spaces, enriching the operator-theoretic understanding of classical matrix operators in complex analysis.
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 $α$-Bloch space $\mathcal{B}^α$ into the logarithmically weighted $\mathcal{B}^α_{\log}$ with $1 <α< 2$, and from the Hardy space $H^{1}$ into the logarithmically weighted Hardy space $H^{1}_{\log}$.