Bourgain–Demeter–Kemp decoupling conjecture for analytic surfaces in three dimensions

Let $\phi:[-1,1]^2\to\mathbb{R}$ be real-analytic and let $\delta>0$. A set $S\subseteq[-1,1]^2$ is $\delta$-flat for $\phi$ when

$$\sup_{u,v\in S}\left|\phi(v)-\phi(u)-\nabla\phi(u)\cdot(v-u)\right|\leq\delta.$$

For $f:\mathbb{R}^3\to\mathbb{C}$ and $S\subseteq\mathbb{R}^2$, let $f_S$ denote the Fourier restriction of $f$ to $S\times\mathbb{R}$, so that

$$\widehat{f_S}(x,y,z)=\widehat f(x,y,z)1_S(x,y).$$

Bourgain–Demeter–Kemp decoupling conjecture. There is a partition $\mathcal P_\delta=\mathcal P_\delta(\phi)$ of $[-1,1]^2$ into $\delta$-flat subsets $S$ such that, for every $f$ whose Fourier transform is supported in the $\delta$-neighbourhood of the graph of $\phi$,

$$\|f\|_{L^4(\mathbb{R}^3)}\leq C_{\phi,\varepsilon}\delta^{-\varepsilon}\#\mathcal P_\delta^{1/4}\left(\sum_{S\in\mathcal P_\delta}\|f_S\|_{L^4(\mathbb{R}^3)}^4\right)^{1/4}$$

for every $\varepsilon>0$. The conjecture proposes a decoupling estimate for arbitrary real-analytic surfaces in $\mathbb{R}^3$, extending the positive-curvature theory to surfaces with vanishing Gaussian curvature; the source does not state whether it has been resolved.