Degree formulas for maps with nonintegrable Jacobian

This paper arose from a discussion sparked between the authors after the lecture of Louis Nirenberg at the Conference in Naples on June 1, 1995. He presented a joint work with Haim Brezis [BN] on the degree theory for VMO (vanishing mean oscillation) mappings f : X → Y between n-dimensional smooth manifolds. Their results include a variety of discontinuous maps. We soon realized that we can contribute to their work by studying some Orlicz– Sobolev classes weaker than W (X,Y ). Our approach relies on new estimates for the Jacobians [IS], [GIM] and most recent improvements [I] concerning nonlinear commutators. Also L-Hodge theory [S], [ISS] plays a crucial role in this paper. Let us begin with the well known formula for the degree of a C-map f : X → Y :