Beltrami operators in the plane

We determine optimal L-p-properties for the solutions of the general nonlin ear elliptic system in the plane of the form f (z) over bar = H(z, f(z)), h is an element of L-p(C), where H is a measurable function satisfying \H(z, omega (1)) - H(z, omega ( 2))\ less than or equal to k\omega (1) - omega (2)\ and k is a constant k < 1. We also establish the precise invertibility and spectral properties in L-p( C) for the operators I - T<mu>, I - muT, and T - mu, where T is the Beurling transform. These operators are basic in the theory of quasi-conformal mappings and in linear and nonlinear elliptic partial di fferential equations (PDEs) in two dimensions. In particular we prove inver tibility in L-p(C) whenever 1 + \\mu\\(infinity) < p < 1+1/\\mu\\(infinity) . We also prove related results with applications to the regularity of weakly quasiconformal mappings.