Discretized Carleson Measures For Weighted Besov Spaces

Carleson measures for the weighted holomorphic Besov spaces Bpα(D) on theunit disc in C are characterized in terms of a discrete tree condition on the asso-ciated Bergman tree T. A martingale-like analogue of the weighted holomorphicBesov space on the unit disc is provided by weighted holomorphic Besov spaces onBergman trees. A necessary and su?cient condition is obtained for a measureμto be Carleson measure from Bpα(D) to Lq(μ). This extends the recent result of N.Arcozzi, R. Rochberg and E. Sawyer to the weighted, di?erent indices, and higherderivative cases.