Interpolation theorem for the $p$-harmonic transform

Luigi D'Onofrio¹ ; Tadeusz Iwaniec²

¹ Dipartimento di Matematica e Applicazioni “R. Caccioppoli” Complesso Universitario Monte S. Angelo 80126, Napoli, Italy
² Department of Mathematics Syracuse University Syracuse, NY 13244, U.S.A.

Studia Mathematica, Tome 159 (2003) no. 3, pp. 373-390

Résumé

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.

Détail

DOI : 10.4064/sm159-3-3

Keywords: establish interpolation theorem class nonlinear operators lebesgue spaces scr mathbb arising naturally study elliptic pdes prototype those pdes second order p harmonic equation mathop div nolimits nabla p nabla mathop div nolimits mathfrak example p harmonic transform essentially inverse mathop div nolimits nabla p nabla every vector field mathfrak scr mathbb mathbb operator scr assigns gradient solution scr mathfrak nabla scr mathbb mathbb core matter beyond natural domain definition operator because nonlinearity arguments require substantial innovations compared classical interpolation theory riesz thorin marcinkiewicz subject largely motivated recent developments geometric function theory