Summary: The -harmonic transforms are the most natural nonlinear counterparts of the Riesz transforms in . They originate from the study of the -harmonic type equation
where is a given vector field in and is an unknown function of Sobolev class , . The -harmonic transform assigns to the gradient of the solution: . More general PDE’s and the corresponding nonlinear operators are also considered. We investigate the extension and continuity properties of the -harmonic transform beyond its natural domain of definition. In particular, we identify the exponents for which the operator is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution fails when exceeds certain critical value. In case , there is no uniqueness in for any .