Norm-attaining integral operators on analytic function spaces

TL;DR

This paper investigates norm-attaining properties of integral operators induced by bounded analytic functions on various analytic function spaces, revealing conditions under which these operators attain their norms.

Contribution

It provides new results on when integral operators attain their norms on Bloch, Dirichlet, and BMOA spaces, including cases where they do not.

Findings

$S_g$ attains its norm on Bloch space and BMOA for any $g$.

$S_g$ does not attain its norm on the Dirichlet space for non-constant $g$.

Results for $S_g$ on the little Bloch space and for $T_g$ from Dirichlet to Bergman space.

Abstract

Any bounded analytic function $g$ induces a bounded integral operator $S_g$ on the Bloch space, the Dirichlet space and $BMOA$ respectively. $S_g$ attains its norm on the Bloch space and $BMOA$ for any $g$, but does not attain its norm on the Dirichlet space for non-constant $g$. Some results are also obtained for $S_g$ on the little Bloch space, and for another integral operator $T_g$ from the Dirichlet space to the Bergman space.