LAUSR

Abstract We consider the Cauchy problem for the Hartree equation in space dimension d ≥ 2 . We assume that the interaction potential V ( x ) is short range. More precisely, we consider the case where V belongs to the weak L d / σ space with 1 σ d . We prove that if 2 ≤ σ d (resp. 1 σ 2 ), the initial data ϕ is small in the sense of the homogeneous Sobolev space H ˙ σ / 2 − 1 (resp. the homogeneous weighted Sobolev space F H ˙ 1 − σ / 2 ) and the Fourier transform F ϕ satisfies a real-analytic condition, then the corresponding solution u ( t ) is also real-analytic for any t ≠ 0 . We remark that no H ˙ σ / 2 − 1 (resp. F H ˙ 1 − σ / 2 ) smallness condition is imposed on first and higher order partial derivatives of F ϕ when 2 ≤ σ d (resp. 1 σ 2 ).