The $(n,k)$ maximal-function conjecture

Let $1\leq k<n$, let $G(n,k)$ be the Grassmannian of $k$-dimensional subspaces of $\mathbb{R}^n$, and let $f\in L^1_{\operatorname{loc}}(\mathbb{R}^n)$. For $0<\delta\ll1$, define

$$T_{n,k,\delta}f(\pi)=\sup_{a\in\mathbb{R}^n}\frac{1}{|P_{\delta}(a,\pi)|}\int_{P_{\delta}(a,\pi)}|f(x)|\,dx,$$

where $P_{\delta}(a,\pi)$ is the $\delta$-neighborhood of the $k$-dimensional unit cube parallel to $\pi$ and centered at $a$. Let $d\pi$ be the rotationally invariant probability measure on $G(n,k)$.

$(n,k)$ maximal-function conjecture. For all $\epsilon>0$ and $0<\delta\ll1$,

$$\lVert T_{n,k,\delta}f\rVert_{L^p(G(n,k),d\pi)}\leq C_{\epsilon}\delta^{\frac{n}{p}-k-\epsilon}\lVert f\rVert_{L^p(\mathbb{R}^n,dx)}$$

for $1\leq p\leq\frac{n}{k}$.

This is the analytic counterpart of the $(n,k)$-set dimension conjecture and extends the Kakeya maximal-function problem. The source provides no resolution status, so the conjecture is recorded as open.