BMO and Lipschitz approximation by solutions of elliptic equations

Abstract

We consider the problem of qualitative approximation by solutions of a constant coefficients homogeneous elliptic equation in the Lipschitz and BMO norms. Our method of proof is well-known: we find a sufficient condition for the approximation reducing matters to a weak <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>*</mml:mo> </mml:math> spectral synthesis problem in an appropriate Lizorkin-Triebel space. A couple of examples, evolving from one due to Hedberg, show that our conditions are sharp.