On linear elliptic equations with drift terms in critical weak spaces

We study the Dirichlet problem for a second-order linear elliptic equation in a bounded smooth domain [Formula: see text] in [Formula: see text], [Formula: see text], with the drift [Formula: see text] belonging to the critical weak space [Formula: see text]. We decompose the drift [Formula: see text] in which div [Formula: see text] and [Formula: see text] is small only in a small scale quasi-norm of [Formula: see text]. Under this new smallness condition, we prove existence, uniqueness, and regularity estimates of weak solutions to the problem and its dual. Hölder regularity and derivative estimates of weak solutions to the dual problem are also established. As a result, we prove uniqueness of very weak solutions slightly below the threshold. When [Formula: see text], our results recover those by Kim and Tsai in [Existence, uniqueness, and regularity results for elliptic equations with drift terms in critical weak spaces, SIAM J. Math. Anal. 52(2) (2020) 1146–1191]. Due to the new small scale quasi-norm, our results are new even when [Formula: see text].