On the energy image density conjecture of Bouleau and Hirsch

We affirmatively resolve the energy image density conjecture of Bouleau and Hirsch (1986). This conjecture generalizes a foundational result in Malliavin calculus: the non-degeneracy of the Malliavin matrix of a random variable implies absolute continuity of its law. This property is a key step in Malliavin's probabilistic proof of Hörmander’s hypoellipticity theorem. Going beyond the original framework of Dirichlet structures, we establish the energy image density property in a unified setting that includes classical Dirichlet forms, Sobolev spaces defined via upper gradients, and self-similar energies on fractals. As an application of independent interest, we show that the martingale dimension of a diffusion satisfying a sub-Gaussian heat kernel estimate is finite. As another application, we provide a new proof of Cheeger’s conjecture on the Hausdorff dimension of images of differentiability charts in PI spaces. This is joint work with Sylvester Eriksson-Bique (University of Jyväskylä).