AI 中文总结
研究赫尔德不等式中亏缺，核心方法是建立其精确表示公式，将亏缺表示为自然指数插值上的累积方差，主要贡献是揭示不等式与对数划分函数凸性的关系及提供亏缺的内在解释，还给出研究泛函不等式的新框架。
AI 中文摘要
我们建立了赫尔德不等式中亏缺的精确表示公式。并非通过辅助量来估计亏缺，而是表明它可表示为连接两个端点密度的自然指数插值上的累积方差。更确切地说，对数赫尔德亏缺表示为对数密度比的方差的积分，由与一维插值参数相关的格林核加权。此恒等式揭示赫尔德不等式是对数划分函数凸性的结果，并将亏缺解释为插值能量。该表示为通过方差恒等式研究泛函不等式提供了更广泛框架。
英文摘要
We establish an exact representation formula for the deficit in Hölder's inequality. Rather than estimating the deficit by auxiliary quantities, we show that it can be represented as an accumulated variance along the natural exponential interpolation connecting the two endpoint densities. More precisely, the logarithmic Hölder deficit is expressed as the integral of the variance of the logarithmic density ratio weighted by the Green kernel associated with the one-dimensional interpolation parameter. This identity reveals that Hölder's inequality is a consequence of the convexity of a logarithmic partition function and provides an intrinsic interpretation of the deficit as an interpolation energy. The representation suggests a broader framework for studying functional inequalities through variance identities.
