Norms of the Hilbert Matrix Operator on Some Analytic Function Spaces

Abstract

For $\mathcal{H}:{\mathcal{B}}^{\alpha }\to {\mathcal{B}}^{\alpha }$ with $1<\alpha <2$, we obtain an exact norm representation. In particular, the norm is given by an explicit Gamma-function formula for $1<\alpha \le 3/2$, whereas for $3/2<\alpha <2$ it strictly exceeds the corresponding Gamma-function expression. For $\mathcal{H}:{\mathcal{B}}^{\alpha }\to {\mathcal{B}}_{\log }^{\alpha }$, we determine the exact norm: it is equal to $3/2$ for $1<\alpha \le 4/3$ and is given by an explicit Gamma-function formula for $4/3<\alpha <2$. We also determine the exact norm of $\mathcal{H}:H^1\to H_{\log }^1$ and show that it is equal to $\pi$. Finally, for $0<\alpha <1$, we determine the exact norm of $\mathcal{H}:H_{\alpha }^{\infty }\to {\mathcal{B}}^{\alpha +1}$, which is given by an explicit Gamma-function formula.

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