Interpolation theorem for the p-harmonic transform

We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces ℒ^s (ℝⁿ) arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation div|∇|u^p-2∇u = div f. In this example the p-harmonic transform is essentially inverse to div(|∇|^p-2∇). To every vector field f ∈ ℒ ^q (ℝⁿ, ℝⁿ) our operator ℋ _p assigns the gradient of the solution, ℋ_pf = ∇u ∈ ^p(ℝⁿ, ℝⁿ). The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our arguments require substantial innovations as compared with the classical interpolation theory of Riesz, Thorin and Marcinkiewicz. The subject is largely motivated by recent developments in geometric function theory.