On functions of bounded mean oscillation with bounded negative part

Let \(b\) $b$ be a locally integrable function and \(\mathfrak{M}\) $\mathfrak{M}$ be the bilinear maximal function \(\mathfrak{M}(f,g)(x)=\sup_{Q\ni x}\frac{1}{|Q|}\int_{Q}|f(y)g(2x-y)|dy.\) $$\mathfrak{M}(f,g)(x)=\underset{Q\ni x}{sup}\frac{1}{|Q|}{\int }_Q|f(y)g(2x-y)|dy.$$ In this paper, characterization of the BMO function in terms of commutator \(\mathfrak{M}^{(1)}_{b}\) ${\mathfrak{M}}_b^{(1)}$ is established. Also, we obtain the necessary and sufficient conditions for the boundedness of the commutator \([b, \mathfrak{M}]_{1}\) ${[b,\mathfrak{M}]}_1$ . Moreover, some new characterizations of Lipschitz and non-negative Lipschitz functions are obtained.