Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime

We study a multidimensional stochastic differential equation with additive noise: $$ d X_t=b(t, X_t) dt +d \xi_t, $$ where the drift $b$ is integrable in space and time, and $\xi$ is either a fractional Brownian motion or an $\alpha$-stable process. We show weak existence of solutions to this equation under the optimal condition on integrability indices of $b$, going beyond the subcritical Krylov-R\"ockner (Prodi-Serrin-Ladyzhenskaya) regime. This extends the recent results of Krylov (2020) to the fractional Brownian and L\'evy cases. We also construct a counterexample to demonstrate the optimality of this condition. Our methods are built upon a version of the stochastic sewing lemma of L\^e and the John--Nirenberg inequality.