Given a mapping in the Sobolev space $f\in W^{2,n}_{\rm loc}(\mathbb{R}^n,\mathbb{R}^n)$ I would like to know what is the Sobolev regularity of the Jacobian $J_f=\operatorname{det} Df$.

It is well known and easy to prove that if $u,v\in W^{1,p}\cap L^\infty(\mathbb{R}^n)$, then $uv\in W^{1,p}\cap L^\infty$. Indeed, product of a bounded and an $L^p$ function is in $L^p$ and the same argument applies to the derivatives $\partial_i(uv)=(\partial_i u)v+v\partial_i\in L^p$. Now if $u\in W^{1,n}$ than $u$ has very high integrability (Trudinger's inequality) so if $u,v\in W^{1,n}$ (no longer bounded), then $uv\in W^{1,n}$ must belong to some Orlicz-Sobolev space slightly larger than $W^{1,n}$. Thus my question is:

Let $u_1,\ldots,u_n \in W^{1,n}(B^n(0,1))$. Find an optimal (or close to optimal) Orlicz-Sobolev space $W^{1,P}$ for some Young function $P$ such that $u_1\cdot\ldots\cdot u_n\in W^{1,P}$.

In fact I would like to know if one can find $P$ so that it satisfies the so called divergence condition: $$ \int_0^1 \frac{P(t)}{t^{n+1}}\, dt =\infty. $$ is satisfied.

Since the derivatives of $f\in W^{2,n}(\mathbb{R}^n,\mathbb{R}^n)$ belong to $W^{1,n}$ such a result will imply that $J_f=\det Df\in W^{1,P}.$