Beltrami operators in the plane

We determine optimal Lp-properties for the solutions of the general nonlinear elliptic system in the plane of the form f$\overline{z}$=H(z, fz), h∈ Lp(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 Lp(C) for the operators I−Tμ, I−μT, and T−μ, where T is the Beurling transform. These operators are basic in the theory of quasi-conformal mappings and in linear and nonlinear elliptic partial differential equations (PDEs) in two dimensions. In particular, we prove invertibility in Lp(C) whenever 1+||μ||∞