Norm of the Hilbert Matrix Operator on Bloch-Type Spaces

Abstract

We determine the exact norm of the Hilbert matrix operator $\mathcal{H}$ on the $\alpha$-Bloch space ${\mathcal{B}}^{\alpha }$ for $1<\alpha <2$, together with all norm-attaining functions of norm one. For $1<\alpha \le 3/2$, we obtain an explicit formula for the exact norm in terms of the Gamma function. For $3/2<\alpha <2$, we obtain an exact one-dimensional maximization formula and show that the corresponding maximum is attained at an interior point of $(0,1)$. We also determine the exact norm of $\mathcal{H}:{\mathcal{B}}^{\alpha }\to {\mathcal{B}}_{\log }^{\alpha }$. The norm formula changes at the critical parameter $\alpha =4/3$, and the exact norm is obtained for the full range $1<\alpha <2$. The norm-attaining functions are completely characterized for both operators.

21 pages