BMO harmonic approximation in the plane and spectral synthesis for Hardy- Sobolev spaces (Q914890)

Let \(X\subset {\mathbb{C}}\) be compact and let \(f\in VMO(X)\) be harmonic on the interior of X. Our main result states that f is the limit in VMO(X) of a sequence of functions which are harmonic on neighbourhoods of X. As an application of the method of proof we prove a spectral synthesis theorem in the Hardy-Sobolev spaces consisting of functions with second order derivatives in the Hardy space \(H^ 1({\mathbb{R}}^ 2)\). This result, still open for dimensions larger than 2, can be viewed as a limiting case of the spatial synthesis theorem for the classical Sobolev spaces due to Hedberg and Wolff.