Carleson measures and the generalized Campanato spaces of vector-valued martingales

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 ℒp, ϕ(X) and the (p, ϕ)-Carleson measures is investigated. Specifically, it is proved that for q ɛ [2, ∞), the measure dμ := ||dfk||qdℙ ⊗ dm is a (q, ϕ)-Carleson measure on Ω × ℕ for every f ɛ ℒq, ϕ(X) if and only if X has an equivalent norm which is q-uniformly convex; while for p ɛ (1, 2], the measure dμ:= ||dfk||pdℙ ⊗ dm is a (p, ϕ)-Carleson measure on Ω × ℕ implies that f ɛ ℒp, 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.