Symmetrization of plurisubharmonic and convex functions

TL;DR

This paper demonstrates that Schwarz symmetrization reduces Monge-Ampere energy for certain invariant functions, leading to a sharp inequality, and explores limitations of these results in more general domains.

Contribution

It establishes the energy-decreasing property of symmetrization for invariant functions and derives a sharp inequality, extending understanding of symmetrization effects in complex analysis.

Findings

Symmetrization does not increase Monge-Ampere energy for invariant functions.

A sharp Moser-Trudinger inequality is derived for these functions.

Results do not generalize to all balanced domains, only complex ellipsoids.

Abstract

We show that Schwarz symmetrization does not increase the Monge-Ampere energy for $S^1$-invariant plurisubharmonic functions in the ball. As a result we derive a sharp Moser-Trudinger inequality for such functions. We also show that similar results do not hold for general balanced domains except for complex ellipsoids and discuss related questions for convex functions.