Lelong class

Summary

The Lelong class $\mathcal{L}$ is the set of plurisubharmonic functions 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 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.