Subelliptic p -harmonic maps into spheres and the ghost of Hardy spaces

The story begins with the paper of Muller, [59], who — for the sake of an application to nonlinear elasticity — proved that if the Jacobian determinant Ju of a Sobolev map u ∈ W 1,n loc (Rn ,Rn ) is nonnegative, then it belongs locally to L log L. The result is quite intriguing, since a priori Holder inequality implies only that Ju ∈ L1 and one does not suspect any higher integrability. If one does not assume that Ju is nonnegative, then, as Coifman, Lions, Meyer and Semmes [14] have proved, Ju belongs to the local Hardy space H 1 loc. Since a nonnegative function belongs to the local Hardy space if and only if it belongs locally to L log L, the result of Coifman, Lions, Meyer and Semmes generalizes that of Muller. In fact, Coifman, Lions, Meyer and Semmes proved more. Namely, for 1 < p < ∞, the scalar product of a divergence free vector field E ∈ Lp(Rn ,Rn ) and a curl free vector field B ∈ Lp/(p−1)(Rn ,Rn ) belongs to the Hardy space H 1. A priori, Holder inequality implies integrability of this expression only. Note that the Jacobian is of this form. The discovery that many algebraic expressions involving partial derivatives belong to the Hardy space turned out to be important and widely applicable in nonlinear partial differential equations. This was first shown by Helein, [40], [41], [42]. Let us describe briefly his result. Consider maps u : Bn → S m from the n-dimensional ball to the mdimensional sphere such that the p-energy of u , given by the functional