In this talk, we study the Corona problem for the Banach algebra \(H^\infty (\mathbb {D}^n)\) of bounded holomorphic functions on the polydisk \(\mathbb {D}^n \subset \mathbb {C}^n \). In this setting, the Corona problem ask whether the polydisk \( \mathbb {D}^n \) is dense in the Gelfand topology in the maximal ideal space of \( H^\infty (\mathbb {D}^n) \).
We discuss certain cases of the Corona problem on the polydisk and present new necessary and sufficient conditions under which the problem can be solved. Our method is based on a new result concerning the solution of special \( \bar {\partial } \) equations on a polydisk.
This is joint work with Alex Brudnyi.