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

Let $\MM$ be the space of finite measures on a Locally compact Polish space, and let $\BG$ be the Gamma distribution on $\MM$ with intensity measure $ν\in \MM$. Let $\nn^{ext}$ be the extrinsic derivative with tangent bundle $T\MM= \cup_{η\in\MM} L^2(η)$, and let $\GA: T\MM\to T\MM$ be measurable such that $\GA_η$ is a positive definite linear operator on $L^2(η)$ for every $η\in \MM$. Moreover, for a measurable function $V$ on $\MM$, let $\d\BG^V= \e^V\d\BG$. We investigate the Poincaré, weak Poincaré and super Poincaré inequalities for the Dirichlet form $$\EE_{\GA,V}(F,G):= \int_\MM \<\GA_η\nn^{ext}F(η), \nn^{ext}G(η)\>_{L^2(η)}\, \d\BG^V(η),$$ which characterize various properties of the associated Markov semigroup. The main results are extended to the space of finite signed measures.