possible exponents which can appear in the interpolation inequality. With Luis Caffarelli and Robert Kohn, Nirenberg would establish corresponding inequalities for certain weighted norms. Caffarelli, Kohn, and Nirenberg's norms were later investigated more fully in notable work by Florin Catrina and Zhi-Qiang Wang. Immediately following Fritz John's introduction of the bounded mean oscillation (BMO) function space in the theory of elasticity, he and Nirenberg gave a further study of the space, proving in particular the "John−Nirenberg inequality," which constrains the size of the set on which a BMO function is far from its average value. Their work, which is an application of the Calderon−Zygmund decomposition, has become a part of the standard mathematical literature. Expositions are contained in standard textbooks on probability, complex analysis, harmonic analysis, Fourier analysis, and partial differential equations. Among other applications, it is particularly fundamental to Jürgen Moser's Harnack inequality and subsequent work. The John−Nirenberg inequality and the more general foundations of the BMO theory were worked out by Nirenberg and Haïm Brézis in the context of maps between Riemannian manifolds. Among other results, they were able to establish that smooth maps which are close in BMO norm have the same topological degree, and hence that degree can be meaningfully defined for mappings of vanishing mean oscillation (VMO). Calculus of variations In the setting of topological vector spaces, Ky Fan developed a minimax theorem with applications in game theory. With Haïm Brezis and Guido Stampacchia, Nirenberg derived results extending both Fan's theory and Stampacchia's generalization of the Lax-Milgram theorem. Their work has applications to the subject of variational inequalities. By adapting the Dirichlet energy, it is standard to recognize solutions of certain wave equations as critical points of functionals. With Brezis and Jean-Michel Coron, Nirenberg found a novel functional whose critical points can be directly used to construct solutions of wave equations. They were able to apply the mountain pass theorem to their new functional, thereby establishing the existence of periodic solutions of certain wave equations, extending a result of Paul Rabinowitz. Part of their work involved small extensions of the standard mountain pass theorem and Palais-Smale condition, which have become standard in textbooks. In 1991, Brezis and Nirenberg showed how Ekeland's variational principle could be applied to extend the mountain pass theorem, with the effect that almost-critical points can be found without requiring the Palais−Smale condition. A fundamental contribution of Brezis and Nirenberg to critical