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Abstract It is the purpose of this work to study the Choquard equation $$\begin{aligned} i\dot{u}-(-\Delta )^s u=\pm |x|^{\gamma }(I_\alpha *|\cdot |^\gamma |u|^p)|u|^{p-2}u \end{aligned}$$ i u ˙ - ( - Δ ) s u = ± | x | γ ( I α ∗ | · | γ | u | p ) | u | p - 2 u in the space $$\dot{H}^s\cap \dot{H}^{s_c}$$ H ˙ s ∩ H ˙ s c , where $$0<s_c<s$$ 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 $$s=1$$ s = 1 and the second one is the fractional regime $$0<s<1$$ 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 $$L^\infty _{T^*}(\dot{H}^{s_c})$$ 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 $$L^2$$ L 2 . This note gives a complementary to the previous works about the same problem in the energy space $$H^1$$ H 1 .