On the energy image density conjecture of Bouleau and Hirsch

We affirmatively resolve the energy image density conjecture of Bouleau and Hirsch (1986). Beyond the original framework of Dirichlet structures, we establish the energy image density property in several related settings. In particular, we formulate a version of the property that encompasses strongly local, regular Dirichlet forms, Sobolev spaces defined via upper gradients, and self-similar energies on fractals, thereby unifying these under a single framework. As applications, we prove the finiteness of the martingale dimension for diffusions satisfying sub-Gaussian heat kernel bounds, and we obtain a new proof of a conjecture of Cheeger concerning the Hausdorff dimension of the images of differentiability charts in PI spaces. The proof of the energy image density property is based on a structure theorem for measures and normal currents in $\mathbb{R}^n$ due to De Philippis--Rindler, together with the notions of decomposability bundles due to Alberti--Marchese and cone null sets due to Alberti--Cs\"ornyei--Preiss and Bate.