On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion

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.