7 problems

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

Let $(X,d,m,\mathcal E,\mathcal F)$ be an MMD space. Suppose it satisfies sub-Gaussian heat kernel estimates with volume growth exponent $\alpha$ and walk dimension $\beta$, and le…

Let $(X,d,m,\mathcal E,\mathcal F)$ be an MMD space, meaning a metric measure Dirichlet space equipped with the associated diffusion. Suppose it satisfies sub-Gaussian heat kernel…

Let $\mathcal M_{z,z';\theta}$ be the boundary z-measure on the Thoma simplex $\Omega$, whose points are written $(\alpha;\beta)$ with nonincreasing nonnegative coordinate sequences…

Morrey embedding conjecture. For the Sierpinski carpet,

Morrey embedding conjecture. On nested fractals, $\delta_{\mathcal{E}}=1$ and, for every $p>1$, there exists $C>0$ such that

Energy measure singularity dichotomy. Then $\Gamma(f,f) \perp m$ for all $f \in \mathcal{F}$. The conjecture predicts singularity of every energy measure under the stated structura…