Norm of the Cesàro operator between some spaces of analytic functions

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