(A) characterization of carleson measures for bergman spaces and its applications

Abstract

Let U be the open unit disk with the normalized Lebesgue measure m and $L_a^p = L^p\bigcap H(U)$. For a positive measure μ on U and p ＞ 1, there exists a constant C satisfying $\int_U\mid{f}\mid^pd\mu\le C \int_U\mid{f}\mid^pdm for all f\in L^p_a$ if and only if μ is a Carlson measure. For 0 ＜ q ＜ p, Luecking found a necessary and sufficient condition for there to exist a constant C satisfying $\left(\int_U\mid f \mid^q d\mu \right)^{1/q} \le C \left( \int_U \mid f \mid^p dm \right)^{1/p} for all f \in L^p_a$ In this paper we generalized this to the higher dimensional spaces and found some applications.