Abstract: In dimension 2, we introduce a distributional Jacobian determinant   for the nonlinear complex gradient $V_\beta(Dv)$ of a function $v\in  W^{1,2 }_\loc$ with $\beta |Dv|^{1+\beta}\in W^{1,2}_\loc$, where $\beta>-1$.  This is new when $\beta\ne0$. 

     Given any planar $\infty$-harmonic function $u$, we show that such distributional Jacobian determinant $\det DV_\beta(Du)$ is a nonnegative Radon measure with some quantitative local lower and upper bounds.

    Denoting by $u_p$ the $p$-harmonic function having the same nonconstant boundary condition as $u$, we show that $\det DV_\beta(Du_p) \to \det DV_\beta(Du)$ as $p\to\infty$ in the weak-$\star$ sense in the space of Radon measure.

Recall that $V_\beta(Du_p)$ is always quasiregular mappings, but $V_\beta(Du)$ is not in general.

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