On the distributional Jacobian of maps from SN into SN in fractional Sobolev and Hölder spaces

H. Brezis and L. Nirenberg proved that if (gk) ⊂ C(S , S ) and g ∈ C(S , S ) (N ≥ 1) are such that gk → g in BMO(S ), then deg gk → deg g. On the other hand, if g ∈ C(S , S ), then Kronecker’s formula asserts that deg g = 1 |SN | ∫ SN det(∇g) dσ. Consequently, ∫ SN det(∇gk) dσ converges to ∫ SN det(∇g) dσ provided gk → g in BMO(S N ). In the same spirit, we consider the quantity J(g, ψ) := ∫ SN ψ det(∇g) dσ, for all ψ ∈ C(S ,R) and study the convergence of J(gk, ψ). In particular, we prove that J(gk, ψ) converges to J(g, ψ) for any ψ ∈ C(S ,R) if gk converges to g in C(S ) for some α > N−1 N . Surprisingly, this result is “optimal” when N > 1. In the case N = 1 we prove that if gk → g almost everywhere and lim supk→∞ |gk − g|BMO is sufficiently small, then J(gk, ψ) → J(g, ψ) for any ψ ∈ C(S,R). We also establish bounds for J(g, ψ) which are motivated by the works of J. Bourgain, H. Brezis, and H.-M. Nguyen and H.-M. Nguyen. We pay special attention to the case N = 1.