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

Joan Manuel Verdera Melenchón

Vol. 159 (1993), No. 2, 379–396

DOI: 10.2140/pjm.1993.159.379

Abstract

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 f 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.

Mathematical Subject Classification 2000

Primary: 30E10

Secondary: 35N99

Milestones

Received: 5 August 1991

Revised: 16 March 1992

Published: 1 June 1993