\((A_ 2)\)-conditions and Carleson inequalities in Bergman spaces (Q1919926)

Let \(\nu\) and \(\mu\) be finite positive Borel measures on the open unit disk \(D\). The authors are interested in a necessary and sufficient condition in order that an inequality \[ \int_D |f|^2 d\nu\leq C \int_D |f|^2 d\mu \] is satisfied for all analytic polynomials \(f\). The main result of the paper states that the inequality is satisfied for all analytic polynomials \(f\) if and only if \(\widehat\nu_r(a)\leq \gamma \widehat\mu_r(a)\) for all \(a\in D\) when \(d\mu= udm\), and \(u\) satisfies an \((A_2)_\partial\)-condition. The \((A_2)_\partial\)-condition is defined by the authors in the paper and several examples concerning this condition are presented.