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arXiv · 2208.11238

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

Alexander Brudnyi
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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.

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Alexander Brudnyi. 2022-08-24. $L^\infty$ Estimates for the Banach-valued $\bar\partial$ problem in a Disk. https://arxiv.org/abs/2208.11238

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