We prove uniform Hölder regularity estimates for a transport-diffusion equation with a fractional diffusion operator and a general advection field in of bounded mean oscillation, as long as the order of the diffusion dominates the transport term at small scales; our only requirement is the smallness of the negative part of the divergence in some critical Lebesgue space. In comparison to a celebrated result by Silvestre, our advection field does not need to be bounded. A similar result can be obtained in the supercritical case if the advection field is Hölder continuous. Our proof is inspired by Kiselev and Nazarov and is based on the dual evolution technique. The idea is to propagate an atom property (i.e., localisation and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.

我们证明了具有分数扩散算子和有界平均振荡的一般平流场的传输扩散方程的均匀 Hölder 正则估计，只要扩散的阶数在小尺度上支配传输项；我们唯一的要求是在一些关键的勒贝格空间中，散度的负部分很小。与 Silvestre 的著名结果相比，我们的平流场不需要有界。如果对流场是 Hölder 连续的，则在超临界情况下可以获得类似的结果。我们的证明受到 Kiselev 和 Nazarov 的启发，并基于双重进化技术。这个想法是在与分数扩散算子耦合时，在对偶守恒定律下传播原子性质（即，勒贝格空间中的局部化和可积性）。