Cartan Covers and Doubling Bernstein Type Inequalities on Analytic Subsets of $\mathbb{C}^2$

TL;DR

This paper establishes a version of doubling Bernstein inequalities for analytic functions on subsets of ^2, including singular points, by developing a Cartan estimate for maps in ^2, advancing understanding of analytic set behavior.

Contribution

It introduces a new Cartan estimate for maps in ^2 and applies it to prove Bernstein inequalities on analytic subsets, including singularities.

Findings

Proved a doubling Bernstein inequality for analytic subsets of ^2.

Established a Cartan estimate for maps in ^2.

Extended inequalities to include singular points on the analytic set.

Abstract

We prove a version of the doubling Bernstein inequalities for the trace of an analytic function of two variables on an analytic subset of $\mathbb{C}^2$. The estimate applies to the whole analytic set in question including its singular points. The proof relies on a version of the Cartan estimate for maps in $\mathbb{C}^2$ which we establish in this work.