Bouleau–Hirsch energy image density conjecture for Dirichlet structures

Let $(X,\mathcal{X},\mu,\mathcal{E},\mathcal{F})$ be a Dirichlet structure with associated carré du champ operator $\gamma: \mathcal{F} \times \mathcal{F} \to L^1(\mu)$. For $n \in \mathbb{N}$ and $f=(f_1,\ldots,f_n)\in\mathcal{F}^n$, write $\gamma(f)=(\gamma(f_i,f_j))_{i,j}$ and let $\mathcal{L}_n$ denote Lebesgue measure on $\mathbb{R}^n$. The structure has the energy image density property when

$$f_*(\mathds{1}_{\{\det(\gamma(f))>0\}}\cdot\mu)\ll\mathcal{L}_n.$$

Bouleau–Hirsch energy image density conjecture. Every Dirichlet structure satisfies the energy image density property.

The property generalizes Malliavin's non-degeneracy criterion for absolute continuity and is the central conjecture addressed by the paper. According to the abstract, the conjecture is affirmatively resolved, including extensions to strongly local regular Dirichlet forms, Sobolev spaces defined via upper gradients, and self-similar energies on fractals.