The present paper is concerned with Lp-theory of the uniformly elliptic differential operator ℒu = Σij∂/∂Cursive Greek chii(ai,j (Cursive Greek chi)∂u/∂Cursive Greek chij) in ℝn with coefficients of vanishing mean oscillation. Recent estimates for the Riesz transform combined with Fredholm index theory enable us to establish invertibility of the map ℒ: W1,p(ℝn) → W-1,p(ℝn), for every 1< p < ∞. As a side benefit, we obtain the existence and uniqueness theorem for the equation ℒu = μ with a signed measure in the right hand side. Within the framework of quasiconformal mappings we give a fairly general method of constructing solutions to the homogeneous equation ℒu = 0.
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