A note on weak existence for singular SDEs

Recently Krylov [N. V. Krylov, On time inhomogeneous stochastic Itô equations with drift in Ld+1, Ukraïn. Mat. Zh. 72 (2020) 1232-1253] established weak existence of solutions to SDEs for integrable drifts in mixed Lebesgue spaces, whose exponents satisfy the condition 1/q + d/p ≤ 1, thus going below the celebrated Ladyzhenskaya-Prodi-Serrin condition. We present here a variant of such result, whose proof relies on an alternative technique, based on a partial Zvonkin transform; this allows for drifts with growth at infinity and/or in uniformly local Lebesgue spaces.