Interpolation theorem for the $p$-harmonic transform
Volume 159 / 2003
Abstract
We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces ${\scr L}^s({\mathbb R}^n)$ arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order $p$-harmonic equation $\mathop {\rm div}\nolimits |\nabla u|^{p-2}\nabla u=\mathop {\rm div}\nolimits {{\mathfrak f}}.$ In this example the $p$-harmonic transform is essentially inverse to $\mathop {\rm div}\nolimits (|\nabla |^{p-2}\nabla ) $. To every vector field ${{\mathfrak f}} \in {\scr L}^q({ \mathbb R}^n,{ \mathbb R}^n)$ our operator ${\scr H}_p$ assigns the gradient of the solution, ${\scr H}_p{{\mathfrak f}}= \nabla u \in {\scr L}^p ({ \mathbb R}^n,{ \mathbb R}^n).$ 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.
