最优传输与余维数大于1的ABP方法
Optimal Transport and the ABP Method in Higher Codimension
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中文总结 AI 辅助
该研究证明浸入子流形ABP接触集的逆最优传输原理，得到余维数1至4的尖锐椭球型Michael–Simon–Sobolev不等式，此前余维数3、4的尖锐常数未知。
中文摘要 AI 辅助
我们证明了浸入子流形的ABP接触集的逆最优传输原理：子流形上的一个固定函数，对有界凸目标上的每个概率密度，确定一个源测度和一个最优计划。对于均匀椭球目标，对子流形的 disintegration（分解）得到条件测度，其逐纤维L^∞范数满足尖锐加权平均估计，常数为ωₙ/ωₙ₊ₘ，适用于余维数1≤m≤4。这在相同范围内得到尖锐椭球型Michael–Simon–Sobolev不等式，包括其等号情形和对称化结果。据我们所知，即使在欧氏情形下，余维数3和4的尖锐常数此前也未知。
英文摘要
We prove an inverse optimal-transport principle for the ABP contact set of animmersed submanifold. A fixed function on the submanifold determines, for every probability density on a bounded convex target, a source measure and an optimal plan. For a uniform ellipsoidal target, disintegration over the submanifold gives conditional measures whose fiberwise $L^\infty$-norms satisfy a sharp weighted average estimate with constant $ω_n/ω_{n+m}$ in codimensions $1\leq m\leq4$. This yields the sharp ellipsoidal Michael--Simon--Sobolev inequality in the same range, including its equality cases and symmetrization consequences. To the best of our knowledge, even in the Euclidean case, the sharp constants in codimensions three and four were previously unknown.
