The sharp Morrey embedding conjecture for nested fractals

Let $X$ be a nested fractal with Hausdorff dimension $d_H$, walk dimension $d_W$, Dirichlet form $\mathcal{E}$, variation functional $\mathbf{Var}_{p,\mathcal{E}}$, and Sobolev space $W^{1,p}(\mathcal{E})$. For $p>1$, define

$$\lambda=(d_W-d_H)\left(1-\frac{1}{p}\right).$$

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

$$\mu\text{-ess}\sup\limits_{x\neq y}\frac{|f(x)-f(y)|}{d(x,y)^\lambda}\leq C\mathbf{Var}_{p,\mathcal{E}}(f)$$

for every $f\in W^{1,p}(\mathcal{E})$. In particular, $\lambda=\frac{\log(5/3)}{\log 2}(1-1/p)$ for the Sierpinski gasket and $\lambda=1-1/p$ for the Vicsek set. The conjecture would sharpen the known bounds for nested fractals, but is not proved in the supplied text.