The p-harmonic transform beyond its natural domain of definition

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 ℒ^q (Ω, ℝⁿ) and u is an unknown function of Sobolev class W₀^1,p(Ω, ℝⁿ), p + q = pq. The p-harmonic transform ℋ_p: ℒ^q(Ω, ℝⁿ) → ℒ^p(Ω, ℝⁿ) as-signs to f the gradient of the solution: ℋ_pf = ∇u ∈ ℒ^p(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 ℋ_p: ℒ ^λq(Ω, ℝⁿ) → ℒ^λp(Ω, ℝⁿ) is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution ∇u ∈ ℒ^λp(Ω, ℝⁿ) fails when λ exceeds certain critical value. In case p = n = dim Ω, there is no uniqueness in W^1,λn(ℝⁿ) for any λ > 1.