$$L^\infty $$ estimates for the Banach-valued $${{\bar{\partial }}}$$-problem in a disk

Abstract

We study the differential equation $$\frac{\partial G}{\partial {{\bar{z}}}}=g$$ with an unbounded Banach-valued Bochner measurable function g on the open unit disk $${\mathbb {D}}\subset {{\mathbb {C}}}$$ . We prove that under some conditions on the growth and essential support of g such equation has a bounded solution given by a continuous linear operator. The obtained results are applicable to the Banach-valued corona problem for the algebra of bounded holomorphic functions on $${\mathbb {D}}$$ with values in a complex commutative unital Banach algebra.