Lax方程与渐近估计李群的弱正则性
The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups
AI总结：
本文研究了无穷维李代数中的Lax方程，推导了积分展开式，并将BCDH公式推广至乘积积分，从而得到了乘积积分的指数映射显式表示，同时引入积分变换证明了非Banach情况下的正则性命题。
摘要：
我们在无穷维李代数的背景下研究了Lax方程。在序列完备的渐近估计背景下讨论了显式解，并在李代数被Milnor意义下的无穷维李群继承的情况下，推导出了一个积分展开式（即在嵌套换子上的迭代Riemann积分之和，并带有修正项）。在Banach李群（以及具有合适正则性性质的李群）的背景下，我们将Baker-Campbell-Dynkin-Hausdorff公式推广到乘积积分（在非Banach情况下需附加幂零假设）。我们将此公式与针对Lax方程获得的结果相结合，推导出了乘积积分用指数映射表示的显式表达式。非Banach情况下的一个重要要素是我们引入的一个积分变换。该变换将连续的李代数值曲线映射为光滑曲线，并使乘积积分保持不变。该变换也被用于证明渐近估计背景下的一项正则性命题。
英文摘要：
We investigate the Lax equation in the context of infinite-dimensional Lie algebras. Explicit solutions are discussed in the sequentially complete asymptotic estimate context, and an integral expansion (sums of iterated Riemann integrals over nested commutators with correction term) is derived for the situation that the Lie algebra is inherited by an infinite-dimensional Lie group in Milnor's sense. In the context of Banach Lie groups (and Lie groups with suitable regularity properties), we generalize the Baker-Campbell-Dynkin-Hausdorff formula to the product integral (with additional nilpotency assumption in the non-Banach case). We combine this formula with the results obtained for the Lax equation to derive an explicit representation of the product integral in terms of the exponential map. An important ingredient in the non-Banach case is an integral transformation that we introduce. This transformation maps continuous Lie algebra-valued curves to smooth ones and leaves the product integral invariant. This transformation is also used to prove a regularity statement in the asymptotic estimate context.
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