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

TL;DR

This paper establishes $L^ abla$ estimates for the Banach-valued $ard$ problem in a disk, providing conditions for bounded solutions and applications to the Banach-valued corona problem.

Contribution

It introduces new $L^ abla$ estimates for the Banach-valued $ard$ problem and constructs bounded solutions via a continuous linear operator.

Findings

Bounded solutions exist under specific growth and support conditions.

Results apply to the Banach-valued corona problem.

Provides a framework for solving $ard$ equations in Banach spaces.

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.