Riesz transforms and elliptic PDEs with VMO coefficients

The present paper is concerned with L^p-theory of the uniformly elliptic differential operator ℒu = Σ_ij∂/∂Cursive Greek chi_i(ai,j (Cursive Greek chi)∂u/∂Cursive Greek chi_j) 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: W^1,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.