# Sharp norm estimates for composition operators and Hilbert-type inequalities

## TL;DR

This paper establishes sharp norm bounds for composition operators on the Hardy space of Dirichlet series, linking these bounds to discrete Hilbert-type inequalities and the Riemann zeta function.

## Contribution

It provides the first sharp upper bounds for these composition operators in a specific parameter range, connecting operator norms to classical inequalities.

## Key findings

- Sharp upper bounds for composition operator norms on  space.
- Connection between operator norms and Hilbert-type inequalities.
- Identification of the critical parameter  related to the Riemann zeta function.

## Abstract

Let $\mathscr{H}^2$ denote the Hardy space of Dirichlet series $f(s) = \sum_{n\geq1} a_n n^{-s}$ with square summable coefficients and suppose that $\varphi$ is a symbol generating a composition operator on $\mathscr{H}^2$ by $\mathscr{C}_\varphi(f) = f \circ \varphi$. Let $\zeta$ denote the Riemann zeta function and $\alpha_0=1.48\ldots$ the unique positive solution of the equation $\alpha\zeta(1+\alpha)=2$. We obtain sharp upper bounds for the norm of $\mathscr{C}_\varphi$ on $\mathscr{H}^2$ when $0<\operatorname{Re}\varphi(+\infty)-1/2 \leq \alpha_0$, by relating such sharp upper bounds to the best constant in a family of discrete Hilbert-type inequalities.