Carleson Measure Characterization of Weighted BMO Associated with a Family of General Sets

Abstract

In this paper, we introduce a weighted Carleson measure \(d\nu _{\mathbb {E}, w}\) associated with the family \(\mathbb {E}\), where \(\mathbb {E}=\{E_r(x)\}_{r\in \mathcal {I}, x\in X}\) is a family of open subsets of a topological space X endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. Using Calderon–Zygmund theory, we show that the weighted BMO associated with the family \(\mathbb {E}\) can be characterized by the weighted Carleson measure \(d\nu _{\mathbb {E}, w}\).