A remark on optimal data spaces for classical solutions of $\bar\partial$

TL;DR

This paper investigates the minimal regularity conditions needed on data to ensure classical $C^1$ solutions to the inhomogeneous Cauchy-Riemann equations in planar domains, advancing understanding of solution regularity.

Contribution

It identifies the least regularity assumptions on data that still guarantee classical solutions to the $ar ext{ extit{ extbf{ extdollar}}}$-equation.

Findings

Determines minimal data regularity for classical solutions

Provides new regularity thresholds for inhomogeneous Cauchy-Riemann equations

Enhances understanding of solution existence criteria

Abstract

We study the minimal regularity required on the datum to guarantee the existence of classical $C^1$ solutions to the inhomogeneous Cauchy-Riemann equations on planar domains.