Sharp norm estimates for composition operators and Hilbert-type inequalities

Abstract

Let H 2 denote the Hardy space of Dirichlet series f ( s ) = ∑ n ⩾ 1 a n n − s with square summable coefficients and suppose that φ is a symbol generating a composition operator on H 2 by C φ ( f ) = f ∘ φ . Let ζ denote the Riemann zeta function and α 0 = 1.48 … the unique positive solution of the equation α ζ ( 1 + α ) = 2 . We obtain sharp upper bounds for the norm of C φ on H 2 when 0 < Re φ ( + ∞ ) − 1 / 2 ⩽ α 0 , by relating such sharp upper bounds to the best constant in a family of discrete Hilbert‐type inequalities.