Optimal Regularity and Control of the Support for the Pullback Equation

Abstract

Given f ,g two C^r,α either symplectic forms or volume forms on a bounded open set Ω ⊂Rⁿ with 0<α<1 and r ≥0, we give natural conditions for the existence of a map ϕ∈Diff^r+1,α(Ω;Ω) satisfying $$ϕ ∗ (g) = f \;\; {\rm in}  \;\; Ω \quad  {\rm\ and \quad supp}  \;\;(ϕ−id)⊂Ω.$$