Norm Attaining Elements of the Ball Algebra \(H^\infty (B_N)\)

Let \(B_N\) $B_N$ be the Euclidean ball of \({\mathbb {C}}^N\) ${\mathbb{C}}^N$ . The space \(H^\infty (B_N)\) $H^{\infty }(B_N)$ of bounded holomorphic functions on \(B_N\) $B_N$ is known to have a predual, denoted by \(G^\infty (B_N)\) $G^{\infty }(B_N)$ . We study the functions in \(H^\infty (B_N)\) $H^{\infty }(B_N)$ that attain their norm as elements of the dual of \(G^\infty (B_N)\) $G^{\infty }(B_N)$ . We also examine similar questions for the polydisc algebra \(H^\infty ({\mathbb {D}}^N)\) $H^{\infty }({\mathbb{D}}^N)$ and for the space of Dirichlet series \( {\mathcal {D}}^\infty ({\mathbb {C}}_+).\) ${\mathcal{D}}^{\infty }({\mathbb{C}}_{+}).$