Integrability for the Jacobian of Orientation Preserving Mappings

Assume F is a map from Ω into Rn where Ω is a bounded domain in Rn such that |Dƒ| ∈ Ln(logL)−s with 0≥ s ≥ 1, i.e., ∫Ω |Dƒ|n[log(1 +|Dƒ|]−s ≤ ∞, then J ∈ L(log L)1−s(K) for any compact subset K ⊂ Ω, where J=det(Dƒ). When s=0 we recover a well-known result of Müller (J. Reine Angew. Math. 412 (1990), 20-34), while the case s = 1 was obtained by Iwaniec and Sbordone (Arch. Rational Mech. Anal., to appear)