The aim of this paper is to study the Lelong number, the integrability index and the Monge–Ampère mass at the origin of an $S^1$-invariant plurisubharmonic function on a balanced domain in $\mathbb{C}^n$ under the Schwarz symmetrization. We prove that $n$ times the integrability index is exactly the Lelong number of the symmetrization, and if the function is further toric with a single pole at the origin, then the Monge–Ampère mass is always decreasing under the symmetrization.

中文翻译：

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勒隆数，蒙格-安培质量和许亚次谐波函数的Schwarz对称化

本文的目的是研究在$ \ mathbb {C} ^ n $的平衡域上，在一个$ S ^ 1 $不变质子次谐波函数的起点处的Lelong数，可积指数和Monge-Ampère质量Schwarz对称化。我们证明，$ n $乘以可积性指数恰好是对称化的勒隆数，并且如果该函数在原点处具有单个极点的情况下进一步复曲面，那么蒙格一安培的质量在对称化下总是在减小。