Bourgain–Spencer regularity conjecture for homogenized coefficients

Let $d$ denote the spatial dimension, let $\boldsymbol a$ be a random coefficient field satisfying suitable mixing conditions, and let $B$ be the convolution operator associated with the averaged solution in Lemma~. A convolution operator $B_0$ on $\operatorname{L}^2(\mathbb R^d)^d$ is required to satisfy

$$B=-\nabla\cdot B_0\nabla.$$

Bourgain–Spencer conjecture. Under suitable mixing conditions on the random coefficient field $\boldsymbol a$, such an operator $B_0$ exists and its Fourier symbol $\hat B_0$ is of Hölder class $C^{2d-\eta}$ at the origin for all $\eta>0$.

This conjecture extends the small-ellipticity-ratio iid result of Bourgain, in its nearly optimal form due to Kim and the third author, to general mixing conditions and beyond the perturbative regime. The statement concerns the regularity needed to define higher-order homogenized coefficients.