  We study a one dimensional dissipative transport equation with nonlocal
velocity and critical dissipation. We consider the Cauchy problem for initial
values with infinite energy. The control we shall use involves some weighted
Lebesgue or Sobolev spaces. More precisely, we consider the familly of weights
given by $w_{\beta}(x)=(1+\vert x \vert^{2})^{-\beta/2}$ where $\beta$ is a
real parameter in $(0,1)$ and we treat the Cauchy problem for the cases
$\theta_{0} \in H^{1/2} (w_{\beta})$ and $\theta_{0} \in H^{1} (w_{\beta})$ for
which we prove global existence results (under smallness assumptions on the
$L^\infty$ norm of $\theta_0$). The key step in the proof of our theorems is
based on the use of two new commutator estimates involving fractional
differential operators and the family of Muckenhoupt weights.
