Norm of the Cesàro operator between some spaces of analytic functions

Abstract

In this paper, we determine the exact norm of the Cesàro operator $\mathcal{C}$ on the Korenblum space $H_{\alpha }^{\infty }$ for $0<\alpha \le \frac{1}{2}$ and on the logarithmically weighted space $H_{\alpha ,\log }^{\infty }$ for $0<\alpha <1$. Moreover, we compute its norm when acting from $H_{\alpha ,\log }^{\infty }$ to $H_{\alpha }^{\infty }$. Finally, we establish lower and upper bounds for the norm of $\mathcal{C}$ on the $\alpha$-Bloch space ${\mathcal{B}}^{\alpha }$ for $\alpha >1$, and from the Hardy space $H^{\infty }$ to ${\mathcal{B}}^{\alpha }$ for $\alpha \ge 1$.