AI 中文总结
本文针对相位紧分层极小集附近的实拉普拉斯积分，建立含适配法向单元分解等的新方法，填补现有定理未覆盖相关几何与分析假设组合的空白。
AI 中文摘要
我们针对积分 $I(z)=\int_N a(x)e^{-z f(x)}\operatorname{dvol}_g(x)$（$z\to\infty$）在相位的紧、可能分层的极小集附近，建立了实拉普拉斯方法。基础输入为适配的法向单元分解（定义到零集），结合分段各向异性伸缩与齐次极限模型。该框架同时允许有限 $C^2$ 分层、Borel 基细化与 $C^1$ 适配图、可测的基依赖单元、相对域及入射层分配、不同部分的不同纤维维数与权重、连续相位与仅可测的振幅，以及由环境强制性或逐单元指数紧性直接控制的极限模型。单元可能仅在加权测度下收敛，或在可积基优函数下逐点收敛。据我们所知，现有实拉普拉斯积分定理均未涵盖这种几何与分析假设的完整组合，代价是适配分解与极限模型需在每个应用中验证其假设。
英文摘要
We develop a real Laplace method for integrals $$ I(z)=\int_N a(x)e^{-z f(x)}\operatorname{dvol}_g(x),\qquad z\to\infty, $$ near a compact, possibly stratified, minimum set of the phase. The basic input is an adapted normal-cell disintegration, defined up to null sets, together with piecewise anisotropic dilations and homogeneous limiting models. The framework simultaneously permits finite $C^2$ stratifications; Borel base refinements and $C^1$ adapted charts; measurable base-dependent cells, relative domains, and incident-stratum assignments; different fiber dimensions and weights on different pieces; a continuous phase and a merely measurable amplitude; and limiting models controlled either by ambient coercivity or directly by cellwise exponential tightness. The cells may converge only in weighted measure, or pointwise under an integrable base majorant. To the best of our knowledge, no existing theorem for real Laplace integrals accommodates this full combination of geometric and analytic assumptions. The tradeoff is that the adapted disintegration and limiting models are supplied hypotheses to be verified in each application.
