Arab. J. Math. https://doi.org/10.1007/s40065-022-00384-y Arabian Journal of Mathematics Tarek Saanouni · Talal Alharbi On the inter-critical inhomogeneous generalized Hartree equation Received: 3 November 2021 / Accepted: 5 May 2022 © The Author(s) 2022 Abstract It is the purpose of this work to study the Choquard equation i ˙ u − (−) s u = ±|x | γ ( I α ∗|·| γ |u | p )|u | p−2 u in the space ˙ H s ∩ ˙ H s c , where 0 < s c < s corresponds to the scale invariant homogeneous Sobolev norm. Here, one considers to two separate cases. The first one is the classical case s = 1 and the second one is the fractional regime 0 < s < 1 with radial data. One tries to develop a local theory using a new adapted sharp Gagliardo–Nirenberg estimate. Moreover, one investigates the concentration of non-global solutions in L ∞ T ∗ ( ˙ H s c ). One needs to deal with the lack of a mass conservation, since the data are not supposed to be in L 2 . This note gives a complementary to the previous works about the same problem in the energy space H 1 . Mathematics Subject Classification 35Q55 1 Introduction This paper is concerned with the Cauchy problem for a Choquard equation  i ˙ u + u = ǫ |x | γ ( I α ∗|·| γ |u | p )|u | p−2 u ; u |t =0 = u 0 , (1.1) where u : R × R N → C, for some integer N ≥ 1, ǫ =±1 refers to the attractive/repulsive regime. The above problem is said inhomogeneous because of the singular quantity |·| γ , where γ< 0. The Riesz potential is defined on R N by I α := Ŵ( N −α 2 ) Ŵ( α 2 )π N 2 2 α |·| N −α , 0 <α< N . T. Saanouni (B ) · T. Alharbi Department of Mathematics, College of science and Arts in Uglat Asugour, Qassim university, Buraydah, Kingdom of Saudi Arabia E-mail: Tarek.saanouni@ipeiem.rnu.tn; t.saanouni@qu.edu.sa T. Alharbi Faculty of Science of Tunis, LR03ES04 Partial Differential Equations and Applications, University of Tunis El Manar, 2092 Tunis, Tunisia E-mail: ta.alharbi@qu.edu.sa 123