Existence, uniqueness and smoothing estimates for spatially homogeneous Landau-Coulomb equation in $H^{-\text{f}12}$ space with polynomial tail

We demonstrate that the spatially homogeneous Landau-Coulomb equation exhibits global existence and uniqueness around the space $H_3^{-\frac{1}{2}}\cap L_7^1\cap L\log L$. Additionally, we furnish several quantitative assessments regarding the smoothing estimates in weighted Sobolev spaces. The new ingredients of the proof lie in the localized techniques in both phase space and frequency space.