Abstract: In this paper, the so-called (p, φ)-Carleson measure is introduced and the relationship between vector-valued martingales in the general Campanato spaces L_p,φ(X) and the (p, φ)-Carleson measures is investigated. Specifically, it is proved that for q ∈[2, ∞), the measure dμ:=||df_k||^qdP ⊗ dm is a (q, φ)-Carleson measure on Ω×N for every f ∈ L_q,φ(X) if and only if X has an equivalent norm which is q-uniformly convex; while for p ∈ (1, 2], the measure dμ:=||df_k||^pdP⊗dm is a (p, φ)-Carleson measure on Ω×N implies that f ∈ L_p,φ(X) if and only if X admits an equivalent norm which is p-uniformly smooth. This result extends an earlier result in the literature from BMO spaces to general Campanato spaces.

Key words: Carleson measures, BMO martingales, generalized Campanato spaces, uniformly convex (smooth) Banach spaces