(A_2)-Conditions and Carleson Inequalities

Let v and µ be finite positive measures on the open unit disk D. We say that v and µ satisfy the ( v ,µ )-Carleson inequality, if there is a constant C > 0 such that 111 dv C D 111 dµ for all analytic polynomials 1 . In this paper, we study the necessary and sufficient condition for the ( v , µ )-Carleson inequality. We establish it when v or µ is an absolutely continuous measure with respect to the Lebesgue area measure which satisfies the (A )-condition. Moreover, many concrete examples of such measures are given