Addendum to: Dacorogna-Moser theorem on the Jacobian determinant equation with control of support

In Dacorogna-Moser theorem on the pullback equation $\varphi^* (g)=f$ between two prescribed volume forms (with the same total volume), control of support of the solutions can be obtained from that of the initial data, while keeping optimal regularity. This result answers a problem implicitly raised on page 14 of Dacorogna-Moser's original article ("On a partial differential equation involving the Jacobian determinant", Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire 7 (1990), 1-26), and fully generalizes the solution to the particular case of $g\equiv 1$ (prescribed Jacobian PDE, $\text{det}\,\nabla\varphi=f$) given in the author's paper "Dacorogna-Moser theorem on the Jacobian determinant equation with control of support", Discrete Cont. Dyn. Syst. 37 (2017), 4071-4089.