Local growth of pluri-subharmonic functions

TL;DR

This paper establishes new two-sided estimates for the local growth of pluri-subharmonic functions using extremal functions, providing simplified proofs of existing inequalities and proposing a conjecture for further bounds based on capacity comparisons.

Contribution

It introduces novel two-bound estimates for pluri-subharmonic functions' growth and offers simplified proofs of classical inequalities, along with a new conjecture relating capacities and extremal functions.

Findings

Derived two-bound estimates for local growth of pluri-subharmonic functions.

Provided simplified proofs of Bernstein doubling inequality and related results.

Proposed a conjecture linking capacities to Siciak extremal functions.

Abstract

We obtain two-bound estimates for the local growth of pluri-subharmonic functions in terms of Siciak and relative extremal functions. As applications, we give simple new proofs of "Bernstein doubling inequality" and the main result in [Alexander Brudnyi, Local inequalities for pluri-subharmonic functions, Annals Math. 149 (1999), No. 2, pp. 511--533]. We propose a conjecture similar to the comparison theorem in [H. Alexander and B. A. Taylor, Comparison of two capacities in $\mathbb{C}^n$, Math. Z. 186 (1984), 407--417], whose validity allows to obtain bounds for the local growth of pluri-subharmonic functions solely in term of the Siciak extremal functions.