Surjective isometries on a Banach space of analytic functions with bounded derivatives

Let \(H(\mathbb D)\) be the linear space of all analytic functions on the open unit disc \(\mathbb D\) and \(H^p(\mathbb D)\) the Hardy space on \(\mathbb D\). The characterization of complex linear isometries on \(\mathcal {S}^p=\{ f\in H(\mathbb D):f'\in H^p(\mathbb D) \}\) was given for \(1 \le p < \infty \) by Novinger and Oberlin in 1985. Here, we characterize surjective, not necessarily linear, isometries on \(\mathcal {S}^\infty \).