Norm of the Ces\`aro operator between some spaces of analytic functions

TL;DR

This paper precisely calculates the norm of the Cesàro operator on various analytic function spaces, including Korenblum, logarithmically weighted, Bloch, and Hardy spaces, providing exact values and bounds.

Contribution

It determines the exact norm of the Cesàro operator on specific analytic spaces and establishes bounds on its norm in other related spaces, extending previous results.

Findings

Exact norm of or Korenblum space $H^\u2218_0$ for $0<500$

Exact norm of or logarithmically weighted space $H^\u2218_{5,07}$ for $0<5<1$

Bounds for the norm of or 5-Bloch space 5>0$ and from Hardy space to 5-Bloch space for 507$

Abstract

In this paper, we determine the exact norm of the Ces\`aro operator $\mathcal{C}$ on the Korenblum space $H^\infty_\alpha$ for $0 < \alpha \leq \frac12$ and on the logarithmically weighted space $H^\infty_{\alpha,\log}$ for $0 < \alpha < 1$. Moreover, we compute its norm when acting from $H^\infty_{\alpha,\log}$ to $H^\infty_\alpha$. 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 $\alpha\geq 1$.