In this paper, the authors obtain continuity of a class linear operators on variable anisotropic Hardy--Lorentz spaces. addition, also that dual space Hardy-Lorentz spaces is BMO-type with exponents. This result still new even when exponent function $p(\cdot)$ $p$.

We study the coupled Hartree system{??u+V1(x)u=?1(|x|?4?u2)u+?(|x|?4?v2)uinRN,??v+V2(x)v=?2(|x|?4?v2)v+?(|x|?4?u2)vinRN, where N?5, ?>max?{?1,?2}?min?{?1,?2}>0, and V1,V2?LN/2(RN)?Lloc?(RN) are nonnegative potentials. This system is critical in sense of Hardy-Littlewood-Sobolev inequality. For with V1=V2=0 we employ moving sphere arguments integral form to classify positive solutions prove uniq...