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We investigate the well-posedness in the generalized Hartree equation $iu_t +\Delta u + (|x|^{-(N-\gamma)} \ast |u|^p)|u|^{p-2}u=0$, $x \in \mathbb{R}^N$,$0<\gamma<N$, for low powers of nonlinearity, $p<2$. We establish the localwell-posedness for a class of data in weighted Sobolev spaces, following ideasof Cazenave and Naumkin [6]. This crucially relies on the boundedness of theRiesz transform in weighted Lebesgue spaces. As a consequence, we obtain aclass of data that exists globally, moreover, scatters in positive time.Furthermore, in the focusing case in the $L^2$-supercritical setting we obtaina subset of locally well-posed data with positive energy, which blows up infinite time.

Well-posedness in weighted spaces for the generalized Hartree equation with $p<2$