Large scale Sobolev inequalities on metric measure spaces and applications
Large scale Sobolev inequalities on metric measure spaces and applications
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度量测度空间上的大规模索博列夫不等式及其应用
DOI：
10.4171/rmi/557
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发表时间：
2007
影响因子：
1.2
通讯作者：
R. Tessera
中科院分区：
文献类型：
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作者：
R. Tessera
对于度量空间上的函数，我们引入了“给定尺度梯度”的概念。这允许我们在给定的尺度上定义Sobolev不等式。我们证明了在足够大的尺度上满足Sobolev不等式在大尺度等价下是不变的，大尺度等价是粗糙等价的度量度量版本。我们证明了对于满足局部庞加莱不等式的黎曼流形，我们的Sobolev不等式的概念在大尺度上等价于它的经典版本。这些概念为研究随机漫步的大时间对角线行为与空间等周性之间的关系提供了一个自然而有效的观点。将我们的主要结果专化到局部紧群上，我们得到了在拟等距条件下，在可调单模紧生的局部紧群之间，对于每1≤p≤∞，lp -等距轮廓是不变的。这种新方法的定性应用是对准传递测度空间X上谱隙存在的非常一般的表征，为理解这一现象提供了一个自然的观点。
For functions on a metric measure space, we introduce a notion of “gradient at a given scale”. This allows us to define Sobolev inequalities at a given scale. We prove that satisfying a Sobolev inequality at a large enough scale is invariant under large-scale equivalence, a metric-measure version of coarse equivalence. We prove that for a Riemmanian manifold satisfying a local Poincare inequality, our notion of Sobolev inequalities at large scale is equivalent to its classical version. These notions provide a natural and efficient point of view to study the relations between the large time on-diagonal behavior of random walks and the isoperimetry of the space. Specializing our main result to locally compact groups, we obtain that the Lp-isoperimetric profile, for every 1 ≤ p ≤ ∞ is invariant under quasi-isometry between amenable unimodular compactly generated locally compact groups. A qualitative application of this new approach is a very general characterization of the existence of a spectral gap on a quasi-transitive measure space X, providing a natural point of view to understand this phenomenon.