$H^{1}$ boundedness of determinants of vector fields

We consider multilinear operators $T(f_{1},\dots , f_{l})$ given by determinants of matrices of the form $(X_{k}f_{j})_{1\le j,k\le l}$, where the $X_{k}$'s are $C^{\infty }$ vector fields on $\mathbb {R}^{n}$. We give conditions on the $X_{k}$'s so that the corresponding operator $T$ maps products of Lebesgue spaces $L^{p_{1}}\times \dots \times L^{p_{l}}$ into some anisotropic space $H^{1}$, when ${\frac {1}{p_{1}}}+\dots +{\frac {1}{p_{l}}}=1$.