Abstract
The p-harmonic transforms are the most natural nonlinear counterparts of the Riesz transforms in R n . They originate from the study of the p-harmonic type equation di v |⊇u| p-2 ⊇u = divf, where f: Ω → Ρ n is a given vector field in L q (Ω, R n ) and u is an unknown function of Sobolev class W 1,p 0 (Ω, R n ), p + q = pq. The p-harmonic transform H p : L q (Ω,R n ) → L p (Ω,R n ) assigns to f the gradient of the solution: H p f = ⊇u ∈ L p (Ω, R n ). 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 A > 1 for which the operator H p : L λq (Ω, R n ) → L λP (Ω, R n ) is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution ⊇u ∈ L λp (Ω, R n ) fails when A exceeds certain critical value. In case p = n = dim Ω, there is no uniqueness in W 1,λn (R n ) for any A > 1.
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