$L^{\mathrm{\infty }}$ Estimates for the Banach-valued $\bar{\mathrm{\partial }}$ problem in a Disk

We study the differential equation $\frac{\mathrm{\partial }G}{\mathrm{\partial }\bar{z}}=g$ 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.