The p-harmonic transform beyond its natural domain of definition

Luigi D'Onofrio, Tadeusz Iwaniec

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16 Scopus citations


The p-harmonic transforms are the most natural nonlinear counterparts of the Riesz transforms in ℝn. They originate from the study of the p-harmonic type equation div|∇u|p-2∇u = divf, where f: Ω → ℝn is a given vector field in ℒq (Ω, ℝn) and u is an unknown function of Sobolev class W01,p(Ω, ℝn), p + q = pq. The p-harmonic transform ℋp: ℒq(Ω, ℝn) → ℒp(Ω, ℝn) as-signs to f the gradient of the solution: ℋpf = ∇u ∈ ℒp(Ω, ℝn). More general PDE's and the corresponding nonlinear operators are also considered. We investigate the extension and continuity properties of the p-harmonic transform beyond its natural domain of definition. In particular, we identify the exponents λ > 1 for which the operator ℋp: ℒ λq(Ω, ℝn) → ℒλp(Ω, ℝn) is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution ∇u ∈ ℒλp(Ω, ℝn) fails when λ exceeds certain critical value. In case p = n = dim Ω, there is no uniqueness in W1,λn(ℝn) for any λ > 1.

Original languageEnglish (US)
Pages (from-to)683-718
Number of pages36
JournalIndiana University Mathematics Journal
Issue number3
StatePublished - 2004


  • Elliptic PDE's
  • p-harmonic transform

ASJC Scopus subject areas

  • General Mathematics


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