Beltrami Operators in the Plane

We determine optimal L-properties for the solutions of the general nonlinear elliptic system in the plane of the form fz = H(z, fz), h ∈ L(C), where H is a measurable function satisfying |H(z,w1) − H(z,w2)| ≤ k|w1 − w2| and k is a constant k < 1. We also establish the precise invertibility and spectral properties in L(C) for the operators I − T μ, I − μT, and T − μ, where T is the Beurling transform. These operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations (PDEs) in two dimensions. In particular, we prove invertibility in L(C) whenever 1+ ‖μ‖∞ < p < 1+ 1/‖μ‖∞. We also prove related results with applications to the regularity of weakly quasiconformal mappings.