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

We prove that the weak Morrey space W Mqpis contained in the Morrey space Mq1pfor 1 ≤ q1<q ≤ p < ∞. As applications, we show that if the commutator [b, T ] is bounded from Lpto Lp,∞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 Mqpto W Mqp. For b belonging to Lipschitz class, we obtain similar results.