Probability measures with either finite Monge–Ampère energy or finite entropy have played a central role in recent developments in Kähler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose Monge–Ampère measures have finite entropy. We show that these potentials belong to the finite energy class $\mathcal{E}^{\frac{n}{n-1}}$, where $n$ denotes the complex dimension, and provide examples showing that this critical exponent is sharp. Our proof relies on refined Moser–Trudinger inequalities for quasi-plurisubharmonic functions.

中文翻译：

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有限熵 vs 有限能量

具有有限 Monge-Ampère 能量或有限熵的概率度量在 Kähler 几何的最新发展中发挥了核心作用。在本笔记中，我们系统研究了准多次谐波势，其 Monge-Ampère 测度具有有限熵。我们证明这些势属于有限能量类 $\mathcal{E}^{\frac{n}{n-1}}$，其中 $n$ 表示复维，并提供示例表明该临界指数是锋利的。我们的证明依赖于准多次谐波函数的精细 Moser-Trudinger 不等式。