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

We prove that the weak Morrey space WM ^p_q is contained in the Morrey space \(M_{{q_1}}^p\) 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 [6, T] is bounded from M ^p_q to WM ^p_q . For b belonging to Lipschitz class, we obtain similar results.