Characterizations of the BMO and Lipschitz Spaces via Commutators on Weak Lebesgue and Morrey Spaces

We prove that the weak Morrey space W M_(q)^(p) is contained in the Morrey space M_(q1)^(p) for 1 ≤ q1<q ≤ p < ∞. As applications, we show that if the commutator [b, T ] is bounded from L^(p) to L^(p,∞) for some p ∈(1, ∞), then b ∈ BMO, where T is a Calderón-Zygmund operator. Also, for 1 < p ≤ q < ∞, b ∈ BMO if and only if [b, T ] is bounded from M_(q)^(p) to WM_(q)^(p). For b belonging to Lipschitz class, we obtain similar results.