Beltrami equations with coefficient in the fractional sobolev space W^θ, 2/θ

© 2016 American Mathematical Society. In this paper, we look at quasiconformal solutions ϕ: ℂ → ℂ of Beltrami equations ∂z ϕ(z) = μ(z) ∂zϕ(z), where μ ∈ L∞(ℂ) is compactly supported on 𝔻, and ||μ||∞ < 1 and belongs to the fractional Sobolev space Wα, 2/α (ℂ). Our main result states that log ∂z ϕ ∈ Wα, 2/α (ℂ) whenever α ≥ 1/2. Our method relies on an n-dimensional result, which asserts the compactness of the commutator (Formula Presented) between the fractional laplacian (−Δ) β/2 and any symbol b ∈ Wβ, n/β (ℝn), provided that 1 < p < n/β.