Carleson measure for analytic Morrey spaces

Let μμ be a positive Borel measure on the unit disc DD. In this note, we show that the inclusion mapping from analytic Morrey spaces L2,λ(D)L2,λ(D) to quadratic tent-type spaces Tλ∞(μ) is bounded (compact) if and only if μμ is a Carleson measure (vanishing Carleson measure). As a byproduct we get a new version of well-known Carleson measure theorem of Hardy spaces. The inverse Carleson measure problem of L2,λ(D)L2,λ(D) is also studied.