Functional Inequalities for Weighted Gamma Distribution on the Space of Finite Measures

Let $\text{MM}$ be the space of finite measures on a Locally compact Polish space, and let $\text{BG}$ be the Gamma distribution on $\text{MM}$ with intensity measure $\nu\in \MM$\mathrm{.}Let$\nn^{ext}$betheextrinsicderivativewithtangentbundle$T\MM= \cup_{\eta\in\MM} L^2(\eta)$,andlet$\GA: T\MM\to T\MM$ be measurable such that ${\text{GA}}_{\eta }$ is a positive definite linear operator on $L^2(\eta )$ for every $\eta \in \text{MM}$. Moreover, for a measurable function $V$ on $\text{MM}$, let $\d\BG^V= \e^V\d\BG$. We investigate the Poincar'e, weak Poincar'e and super Poincar'e inequalities for the Dirichlet form $${\text{EE}}_{\text{GA},V}(F,G):={\int }_{\text{MM}}\text{<}{\text{GA}}_{\eta }{\text{nn}}^{ext}F(\eta ),{\text{nn}}^{ext}G(\eta ){}_{L^2(\eta )}\text{d}{\text{BG}}^V(\eta ),$$ which characterize various properties of the associated Markov semigroup. The main results are extended to the space of finite signed measures.