---
tag: X4ZHQG9995
kind: concept
name: Lelong class
msc: [32U35, 32U15]
sources:
  - url: https://en.wikipedia.org/wiki/Plurisubharmonic_function
---

## Summary

The Lelong class $\mathcal{L}$ is the set of [[plurisubharmonic functions|FF2P7ER3QT]] on $\C^n$ that grow at most like $\log\abs{z}$ at infinity.
It is the analogue, for functions, of the set of polynomials of bounded degree.
For every polynomial $p$ of degree $d \ge 1$, the function $\frac{1}{d} \log\abs{p}$ lies in $\mathcal{L}$.
The [[Siciak extremal function|QBJ9R06GYE]] of a set is defined as an upper envelope of functions of the Lelong class.
The subclass $\mathcal{L}^+$ of functions that also grow at least like $\log\abs{z}$ contains the regularised extremal functions of compact sets that are not pluripolar.

## Explanations

### L1 human

A polynomial of degree $d$ grows like $\abs{z}^d$, so its logarithm divided by $d$ grows like $\log\abs{z}$.
The Lelong class collects all plurisubharmonic functions with this kind of logarithmic growth.
For example, $\log(1 + \abs{z})$ and $\max(\log\abs{z_1}, \log\abs{z_2})$ are in the Lelong class of $\C^2$.
The function $\abs{z_1}$ is plurisubharmonic but grows too fast, so it is not in the class.