Global well-posedness for fractional Hartree equation on modulation spaces and Fourier algebra

We study the Cauchy problem for fractional Schr\"odinger equation with cubic convolution nonlinearity ($i\partial_t u - (-\Delta)^{\frac{\alpha}{2}}u\pm (K\ast |u|^2) u =0$) with Cauchy data in the modulation spaces $M^{p,q}(\mathbb R^{d}).$ For $K(x)= |x|^{-\gamma}$ $ (0< \gamma< \text{min} \{\alpha, d/2\})$, we establish global well-posedness results in $M^{p,q}(\mathbb R^{d}) (1\leq p \leq 2, 1\leq q < 2d/ (d+\gamma))$ when $\alpha =2, d\geq 1$, and with radial Cauchy data when $d\geq 2, \frac{2d}{2d-1}< \alpha < 2. $ Similar results are proven in Fourier algebra $\mathcal{F}L^1(\mathbb R^d) \cap L^2(\mathbb R^d).$