On the uniform approximation problem for the square of the Cauchy-Riemann operator (Q1314996)

Let \(X\) be a compact subset of the complex plane and \(f\) a continuous function on \(X\) satisfying the differential equation \(\overline \partial^ 2f = 0\) in the interior of \(X\). It is unknown whether \(f\) can be uniformly approximated on \(X\) by functions \(g\) satisfying the equation \(\overline \partial^ 2g = 0\) in some neighborhood (depending on \(g)\) of \(X\). The author shows that this is the case under the additional assumption that \(f\) satisfies a Dini-type continuity condition.