We consider a type of nonlocal nonlinear Schrödinger equation, the Calogero-Sutherland DNLS equation.
It turns out that, it is an integrable PDE enjoying a Lax pair structure. Using the integrable tools, we prove the global well-posedness of the Cauchy problem in all the Hardy Sobolev spaces $H^s_+(\mathbb{T})$, $s\geq 0$. A sketch of the proof for extending the flow to the critical regularity $L^2_+(\mathbb{T})$ will be presented.