Nash inequality for Diffusion Processes Associated with Dirichlet Distributions

For any $N\ge 2$ and $\alpha=(\alpha_1,\cdots, \alpha_{N+1})\in (0,\infty)^{N+1}$, let $\mu^{(N)}_{\alpha}$ be the Dirichlet distribution with parameter $\alpha$ on the set $\Delta^{ (N)}:= \{ x \in [0,1]^N:\ \sum_{1\le i\le N}x_i \le 1 \}.$ The multivariate Dirichlet diffusion is associated with the Dirichlet form $${\scr E}_\alpha^{(N)}(f,f):= \sum_{n=1}^N \int_{ \Delta^{(N)}} \bigg(1-\sum_{1\le i\le N}x_i\bigg) x_n(\partial_n f)^2(x)\,\mu^{(N)}_\alpha(d x)$$ with Domain ${\scr D}({\scr E}_\alpha^{(N)})$ being the closure of $C^1(\Delta^{(N)})$. We prove the Nash inequality $$\mu_\alpha^{(N)}(f^2)\le C {\scr E}_\alpha^{(N)}(f,f)^{\frac p{p+1} }\mu_\alpha^{(N)} (|f|)^{\frac 2 {p+1}},\ \ f\in {\scr D}({\scr E}_\alpha^{(N)}), \mu_\alpha^{(N)}(f)=0$$ for some constant $C>0$ and $p= (\alpha_{N+1}-1)^+ +\sum_{i=1}^N 1\lor (2\alpha_i),$ where the constant $p$ is sharp when $\max_{1\le i\le N} \alpha_i \le 1/2$ and $\alpha_{N+1}\ge 1$. This Nash inequality also holds for the corresponding Fleming-Viot process.