标题： 尖锐的定量稳定性对于Prékopa-Leindler和Borell-Brascamp-Lieb不等式 显示英文标题

标题： Sharp Quantitative Stability for the Prékopa-Leindler and Borell-Brascamp-Lieb Inequalities

摘要： Borell-Brascamp-Lieb不等式是Prékopa-Leindler不等式的经典推广，而Prékopa-Leindler不等式本身是Brunn-Minkowski不等式的函数对应形式。 这些不等式的稳定性近年来受到了广泛关注。 尽管在几何设置中取得了重大进展，但Prékopa-Leindler不等式的精确定量稳定性结果仍然难以捉摸，甚至在对数凹函数的特殊情况下也是如此。 在本工作中，我们为这些基础不等式提供了一个统一且确定的稳定性框架。 通过在完全一般的情况下建立Borell-Brascamp-Lieb不等式的最优定量稳定性，我们将Prékopa-Leindler不等式的猜想精确稳定性作为特殊情况加以解决。 我们的方法基于作者最近获得的Brunn-Minkowski不等式的精确稳定性结果。 显示英文摘要

摘要： The Borell-Brascamp-Lieb inequality is a classical extension of the Pr\'ekopa-Leindler inequality, which in turn is a functional counterpart of the Brunn-Minkowski inequality. The stability of these inequalities has received significant attention in recent years. Despite substantial progress in the geometric setting, a sharp quantitative stability result for the Pr\'ekopa-Leindler inequality has remained elusive, even in the special case of log-concave functions. In this work, we provide a unified and definitive stability framework for these foundational inequalities. By establishing the optimal quantitative stability for the Borell-Brascamp-Lieb inequality in full generality, we resolve the conjectured sharp stability for the Pr\'ekopa-Leindler inequality as a particular case. Our approach builds on the recent sharp stability results for the Brunn-Minkowski inequality obtained by the authors.