SDEs with critical time dependent drifts: weak solutions

We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T]; L^{p}(mathbb{R}^d))$ with $d/p+2/q=1$. The weak uniqueness is obtained by solving corresponding Kolmogorovs backward equations in some second order Sobolev spaces, which is analytically interesting in itself.