  We consider a one dimensional nonlocal transport equation and its natural
multi-dimensional analogues. By using a new pointwise inequality for the
Hilbert transform, we give a short proof of a nonlinear inequality first proved
by C\'{o}rdoba, C\'{o}rdoba and Fontelos in 2005. We also prove several new
weighted inequalities for the Hilbert transform and various nonlinear versions.
Some of these results generalize to a related family of nonlocal models.
