Curved commutators in higher dimensions

We complete the characterization of the $L^p$ boundedness of the commutators $[b,H_{\gamma }]$ of pointwise multiplication and the Hilbert transform along monomial curves in any dimension. This fully solve a key point left open by Bongers, Guo, Li and Wick, which was affirmatively answered recently only for dimension two. We achieve this by a novel but elementary argument, avoiding the complicated geometry structure in higher dimensions.