Beltrami operators in the plane

Abstract

We determine optimal L^p-properties for the solutions of the general nonlinear elliptic system in the plane of the form

f_{$\overline{z}$}=H(z, f_z), h∈ L^p(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^p(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 L^p(C) whenever 1+||μ||_∞ <p<1+1/||μ||_∞.

We also prove related results with applications to the regularity of weakly quasiconformal mappings.