Nash inequality for diffusion processes associated with Dirichlet distributions

For any N ⩾ 2 and α = (α1,…, αN+1) ∈ (0, ∞)N+1, let µa(n) be the Dirichlet distribution with parameter α on the set Δ(N):= [x ∈ [0,1]N: Σ1⩽i⩽Nxi ⩽ ]. The multivariate Dirichlet diffusion is associated with the Dirichlet form $${\mathcal E}_\alpha ^{(N)}(f,f): = \sum\limits_{n = 1}^N {\int_{{\Delta ^{(N)}}} {\left( {1 - \sum\limits_{1 \leqslant i \leqslant N} {{x_i}} } \right){x_n}{{({\partial _n}f)}^2}(x)\mu _\alpha ^{(N)}} (dx)}$$ with Domain $${\mathcal D}({\mathcal E}_\alpha ^{(N)})$$ being the closure of C1(Δ(N)). We prove the Nash inequality $$\mu _\alpha ^{(N)}({f^2})C{\mathcal E}_\alpha ^{(N)}{(f,f)^{p/(p + 1)}}\mu _\alpha ^{(N)}{(|f|)^{2/(p + 1)}},\;\;\;f \in {\mathcal D}({\mathcal E}_\alpha ^{(N)}),\mu _\alpha ^{(N)}(f) = 0,$$ for some constant C > 0 and p = (αN+1–1)+ + Σi=1N 1 V (2αi), where the constant p is sharp when max1⩽i⩽Nαi ⩽ 1/2 and αN+1 ⩾ 1. This Nash inequality also holds for the corresponding Fleming-Viot process.