Interpolation theorem for the p-harmonic transform

• Autorzy: Luigi D'Onofrio , Tadeusz Iwaniec
• Języki publikacji: EN

Abstrakty

EN

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, $ℋ_{p}𝔣 = ∇u ∈ ℒ^{p}(ℝⁿ,ℝⁿ)$. 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.

Kategorie tematyczne

• 47B38: Operators on function spaces (general)
• 35J60: Nonlinear elliptic equations
• 41A05: Interpolation

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2003
• Tom: 159
• Numer: 3
• Strony: 373-390

Daty

wydano

2003

Twórcy

autor

Luigi D'Onofrio

• Dipartimento di Matematica, e Applicazioni "R. Caccioppoli", Complesso Universitario Monte S. Angelo, 80126, Napoli, Italy

autor

Tadeusz Iwaniec

• Department of Mathematics, Syracuse University, Syracuse, NY 13244, U.S.A.

Identyfikatory

DOI

10.4064/sm159-3-3