A Sharp Nonlinear Gagliardo–Nirenberg-Type Estimate and Applications to the Regularity of Elliptic Systems

The seminal paper of Coifman et al. (1993) has triggered numerous new applications of Hardy spaces and the space BMO of functions of bounded mean oscillation to nonlinear partial differential equations. These applications include various problems of variational and geometric origin: harmonic and p-harmonic maps, H-systems, wave maps; (see, e.g., Bethuel, 1993; Evans, 1991; Hélein, 1998, 2002; Toro and Wang, 1995) and many others. In a vague sense, a common feature of all those works is that the duality of BMO and the Hardy space 1 are much more subtle than the duality of L and L1. In many instances, the local BMO norm of a function