Norm of the Hilbert matrix operator between some spaces of analytic functions

Abstract

In this paper, we calculate the exact value of the norm of the Hilbert matrix operator $\mathcal{H}$ from the logarithmically weighted Korenblum space $H^\infty_{α,\log}$ into Korenblum space $H^\infty_α$, and from the Hardy space $H^\infty$ to the classical Bloch space $\mathcal{B}$. Furthermore, we compute the precise value of the norm on the logarithmically weighted Korenblum space $H^\infty_{α,\log}$, and obtain both the lower and upper bounds of the norm on $α$-Bloch space $\mathcal{B}^α$. Finally, in the context of mapping from the Korenblum space $H^\infty_α$ to the $(α+1)$-Bloch space $\mathcal{B}^{α+1}$, we establish the norm of $\mathcal{H}$.

Theorems & Definitions (10)

• Theorem 3.1
• Lemma 3.1
• Lemma 3.2
• Theorem 3.2
• Theorem 4.1
• Lemma 5.1
• Theorem 5.1
• Theorem 5.2
• Theorem 6.1
• Theorem 7.1