Derivative bounds and continuity of maximal commutators

We study the regularity properties of maximal commutators acting on $W^{1,1}$-functions. More precisely, let $\mathfrak {M}_{b}$ be the one-dimensional maximal commutator with symbol $b$. Under the condition that $b\in W^{1,1}(\mathbb {R})$ and $\|b’\|_{L