A note on weak existence for singular SDEs

Recently Krylov established weak existence of solutions to SDEs for integrable drifts in mixed Lebesgue spaces, whose exponents satisfy the condition $1\mathrm{/}q+d\mathrm{/}p\le 1$, 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.