The gradient estimate for the critical vector-field exponent

Let $(\sigma_k)_{k\in\mathbb{Z}^2_0}$ be the family of vector fields on $\mathbb{T}^2$ defined, for $\gamma\geq 2$, by

$$\sigma_k(x)={\rm e}^{2\pi {\rm i} k\cdot x}\frac{k^\perp}{|k|^\gamma},\qquad x\in\mathbb{T}^2,\quad k\in\mathbb{Z}^2_0:=\mathbb{Z}^2\setminus\{0\}.$$

Assume that the associated gradient estimate is the estimate referred to as. The gradient-estimate conjecture. The gradient estimate holds when $\gamma=2$ in.

This is the critical case needed to prove the stated $L^2$ convergence of the drift series and hence is relevant to uniqueness for the two-dimensional stochastic Euler equation. The source presents the assertion as a natural problem; no resolution is supplied in the given text.