  We study a 1D transport equation with nonlocal velocity with subcritical or
supercritical dissipation. For all data in the weighted Sobolev space
$H^{k}(w_{\lambda,\kappa}) \cap L^{\infty},$ with $k=\max(0,3/2-\alpha)$ and
$w_{\lambda, \kappa}$ is a given family of Muckenhoupt weights. We prove a
global existence result in the subcritical case $\alpha \in (1,2)$. We also
prove a local existence theorem for large data in $H^{2}(w_{\lambda,
\kappa})\cap L^{\infty}$ in the supercritical case $\alpha \in (0,1)$. The
proofs are based on the use of the weighted Littlewood-Paley theory,
interpolation along with some new commutator estimates.
