Infinite energy solutions for a 1D transport equation with nonlocal velocity

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 $w_{\beta }(x)=(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{-\beta \mathrm{/}2}$ where $\beta$ is a real parameter in $(0,1)$ and we treat the Cauchy problem for the cases ${\theta }_0\in H^{1\mathrm{/}2}(w_{\beta })$ and ${\theta }_0\in H^1(w_{\beta })$ for which we prove global existence results (under smallness assumptions on the $L^{\mathrm{\infty }}$ 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.