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.