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

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.