Assume F is a map from Ω into Rn where Ω is a bounded domain in Rn such that |Df(hook)| ∈ Ln(logL)-s with 0≥ s ≥ 1, i.e., ∫Ω |Df(hook)|n[log(1 +|Df(hook)|]-s ≤ ∞, then J ∈ L(log L)1-s(K) for any compact subset K ⊂ Ω, where J=det(Df(hook)). 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).
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