The Cauchy problem for the Hartree type equation in modulation spaces

We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F(u)=(K∗|u|2)uF(u)=(K∗|u|2)u under a specified condition on potential KK with Cauchy data in modulation spaces Mp,q(Rd)Mp,q(Rd). We establish global well-posedness results in M1,1(Rd)M1,1(Rd) when K(x)=λ|x|−γK(x)=λ|x|−γ(λ∈R,0<γ