Another approach to biting convergence of Jacobians

We give new proof of the theorem of K. Zhang [Z] on biting convergence of Jacobian determinants for mappings of Sobolev class script W sign ^1,n(Ω, ℝⁿ). The novelty of our approach is in using script W sign^1,p-estimates with the exponents 1 ≤ p < n, as developed in [IS1], [IL], [I1]. These rather strong estimates compensate for the lack of equi-integrability. The remaining arguments are fairly elementary. In particular, we are able to dispense with both the Chacon biting lemma and the Dunford-Pettis criterion for weak convergence in ℒ ¹(Ω). We extend the result to the so-called Grand Sobolev setting. Biting convergence of Jacobians for mappings whose cofactor matrices are bounded in ℒ^n/n-1 (ℝⁿ) is also obtained. Possible generalizations to the wedge products of differential forms are discussed.