infinite-dimensional李群的可测正则性
Measurable regularity properties of infinite-dimensional Lie groups
摘要：
我们考虑李群G（可能是无限维的）上的形式为y '（t）=f（t，y（t））的微分方程，其中f：[0，1] x G -> TG是时间依赖的左不变向量场，它对t具有可测的（但不一定是连续的）依赖性.如果G的李代数g中对应于L^1-曲线c：[0，1] -> g的每个f都存在从G的中性元e开始的关于[0，1]的解Evol（c）：=y，并且Evol作为从L^1（[0，1]，g）到C（[0，1]，G）的映射是光滑的，则G称为L^1-正则的.我们证明了所有的Banach-李群都是L^1-正则的，以及有限维李群的所有直接极限和无限维李群的许多进一步的经典例子，如仿紧有限维光滑流形的自同态群。如果李群G是L^1-正则的，则Trotter乘积公式和换位子公式在强意义下在G中成立。如果G仅满足某些较弱的可测正则性（如L^p-正则性），则同样的结论成立，并讨论了这些性质。作为工具，我们研究了向量值Lebesgue空间和向量值绝对连续函数空间的某些非线性映射的可微性。
英文摘要：
We consider differential equations of the form y'(t)=f(t,y(t)) on a (possibly infinite-dimensional) Lie group G, for f : [0,1] x G -> TG a time-dependent left invariant vector field with measurable (but not necessarily continuous) dependence on t. If a solution Evol(c):=y on [0,1] starting at the neutral element e of G exists for each f corresponding to an L^1-curve c : [0,1] -> g in the Lie algebra g of G, and Evol is smooth as a map from L^1([0,1],g) to C([0,1],G), then G is called L^1-regular. We show that all Banach-Lie groups are L^1-regular, as well as all direct limits of finite-dimensional Lie groups and many further classical examples of infinite-dimensional Lie groups, like diffeomorphism groups of paracompact finite-dimensional smooth manifolds. If a Lie group G is L^1-regular, then the Trotter product formula and the commutator formula hold in G, in a strong sense. The same conclusion holds if G merely satisfies certain weaker measurable regularity properties (like L^p-regularity), which are discussed as well. As a tool, we study differentiability properties of certain non-linear mappings to vector-valued Lebesgue spaces and spaces of vector-valued absolutely continuous functions.
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