  In this paper, we provide modulated interaction energy estimates for the
kernel $K(x) = |x|^{-\alpha}$ with $\alpha \in (0,d)$, and its applications to
quantified asymptotic analyses for kinetic equations. The proof relies on a
dimension extension argument for an elliptic operator and its commutator
estimates. For the applications, we first discuss the quantified small inertia
limit of kinetic equation with singular nonlocal interactions. The aggregation
equations with singular interaction kernels are rigorously derived. We also
study the rigorous quantified hydrodynamic limit of the kinetic equation to
derive the isothermal Euler or pressureless Euler system with the nonlocal
singular interactions forces.
