On the uniform approximation problem for the square of the cauchy-riemann operator

Let X be a compact subset of the plane and f a continuous function on X satisfying the equation ∂-2f = 0 in the interior of X. It is unknown whether / can be uniformly approximated on X by functions g satisfying the equation ∂-2g = 0 in some neighbourhood (depending on g) of X. We show that this is the case under the additional assumption that f satisfies a Dini-type continuity condition. © 1993 by Pacific Journal of Mathematics.