Sharp norm estimates for composition operators and Hilbert-type inequalities

Abstract

Let ${\mathcal{H}}^2$ denote the Hardy space of Dirichlet series $f(s)={\sum }_{n\ge 1}{a}_n{n}^{-s}$ with square summable coefficients and suppose that $\phi$ is a symbol generating a composition operator on ${\mathcal{H}}^2$ by ${\mathcal{C}}_{\phi }(f)=f\circ \phi$. Let $\zeta$ denote the Riemann zeta function and ${\alpha }_0=1.48\dots$ the unique positive solution of the equation $\alpha \zeta (1+\alpha )=2$. We obtain sharp upper bounds for the norm of ${\mathcal{C}}_{\phi }$ on ${\mathcal{H}}^2$ when $0<\mathrm{Re}\phi (+\infty )-1/2\le {\alpha }_0$, by relating such sharp upper bounds to the best constant in a family of discrete Hilbert-type inequalities.

This paper has been accepted for publication in Bulletin of the LMS

References in corpus (1)

• An embedding constant for the Hardy space of Dirichlet series

Cited by in corpus (8)

• A mean counting function for Dirichlet series and compact composition operators
• Norms of composition operators on the $H^2$ space of Dirichlet series
• Composition operators on weighted Hilbert spaces of Dirichlet series
• The spectrum of some Hardy kernel matrices
• Idempotent Fourier multipliers acting contractively on $H^p$ spaces
• The best constant in a Hilbert-type inequality
• The spectral density of Hardy kernel matrices
• Topological structure of the space of composition operators on the Hardy space of Dirichlet series