The restricted k-broad l^p-decoupling conjecture

Let $2\leq k\leq n+1$. For a function $F$ on $\mathbb{R}^{n+1}$ with Fourier support in $N_{1/D^2}\Sigma$, let $F_{D,V}$ denote its contribution associated with a $k$-dimensional subspace $V\in\mathcal{V}_k$, let $B\in\mathcal{Q}_{D^2}$ be a dyadic cube in $\mathbb{R}^{n+1}$, and let $B^*$ be the prescribed $D_0$-dilation of $B$. Let $\|F_{D,V}\|_{p,D,B^*}$ denote the $l^p$-decoupling norm with parameter $D$, and let $\|F_{D,V}\|_{BL^p_{k,A,D_0}(B)}$ denote the corresponding $k$-broad norm. Assume that $A$ is an integer satisfying $(\log D)^L\ll A\ll D_0$. Restricted $k$-broad $l^p$-decoupling conjecture. If $p>\frac{2(n+1)}{n}$, then for every $\varepsilon>0$ there is a constant $C_\varepsilon$ such that

$$\|F_{D,V}\|_{BL^p_{k,A,D_0}(B)}\leq C_\varepsilon D^{n-\frac{2(n+1)}{p}+\varepsilon}\|F_{D,V}\|_{p,D,B^*}$$

for sufficiently large dyadic $D$, any such $F$, any dyadic cube $B\in\mathcal{Q}_{D^2}$, any $V\in\mathcal{V}_k$, and any admissible $A$. This would extend the stated theorem to the conjectured lower range of $p$ and would yield improved estimates for the Fourier restriction conjecture; the paper gives no resolution of this claim.