Let $mathbb{M} $ be the space of finite measures on a locally compact Polish space, and let $mathcal{G} $ be the Gamma distribution on $mathbb{M} $ with intensity measure $u in mathbb{M} $. Let $abla ^{ext}$ be the extrinsic derivative with tangent bundle $Tmathbb{M} = cup _{eta in mathbb{M} } L^{2}(eta )$, and let $mathcal{A} : Tmathbb{M} ightarrow Tmathbb{M} $ be measurable such that $mathcal{A} _{eta }$ is a positive definite linear operator on $L^{2}(eta )$ for every $eta in mathbb{M} $. Moreover, for a measurable function $V$ on $mathbb{M} $, let ${mathrm{{d}} }{mathcal{G} }^{V}= {mathrm{{e}} }^{V}{mathrm{{d}} }{mathcal{G} }$. We investigate the Poincaré, weak Poincaré and super Poincaré inequalities for the Dirichlet form [ mathcal{E} _{mathcal{A} ,V}(F,G):= int _{mathbb{M} }langle mathcal{A} _{eta }abla ^{ext}F(eta ), abla ^{ext}G(eta )angle _{L^{2}(eta )}, {mathrm{{d}} }{mathcal{G} }^{V}(eta ), ] which characterize various properties of the associated Markov semigroup. The main results are extended to the space of finite signed measures.
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机译：让$ mathbb {m} $是本地紧凑的波兰空间上有限度量的空间，让$ mathcal {g} $是$ mathbb {m} $上的伽玛分布，强度测量$ nu mathbb {m} $。让$ nabla ^ {ext} $是与切线束$ t mathbb {m} = cup _ { eta in mathbb {m}} l ^ {2}（ eta）$的子项衍生物让$ mathcal {a}：t mathbb {m} lightarrow t mathbb {m} $可衡量，即$ mathcal {a} _ { eta} $是$ l ^ {的正定线性运算符2}（ eta）$每一个$ eta in mathbb {m} $。此外，对于可测量的函数$ v $ v $ mathbb {m} $，让$ { mathrm {{{{d}}} { mathcal {g}} ^ {v} = { mathrm {{e}}} ^ {v} { mathrm {{d}}} { mathcal {g}} $。我们调查Dirichlet Form的Poincaré，弱庞加玲和超级Poincaré不等式 [ Mathcal {e} _ { mathcal {a}，v}（f，g）：= int _ { mathbb {m}}} langle mathcal {a} _ { eta} nabla ^ {ext} f（ eta）， nabla ^ {ext} g（ eta） rangle _ {l ^ {2}（ eta）} ， { mathrm {{{{{{{{g}} ^ {v}（ eta），]，其表征关联的Markov半群的各种属性。主要结果延伸到有限签署措施的空间。
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