H\"older estimates for fractional parabolic equations with critical divergence free drifts

This work focuses on drift-diffusion equations with fractional dissipation $(-\Delta)^{\alpha}$ in the regime $\alpha \in (1/2,1)$. Our main result is an a priori H\"older estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $\beta \in (0,1)$, the $C^{\beta}$ norm of the solution depends only on the size of the drift in critical spaces of the form $L^{q}_{t}(BMO^{-\gamma}_{x})$ with $q>2$ and $\gamma \in (0,2\alpha-1]$, along with the $L^{2}_{x}$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.