  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.
