Dimension-free estimates for discrete maximal functions of q-balls

Let $d\in\mathbb{N}$ and let $B^q(d)\subset\mathbb{R}^d$ be a $q$-ball. The discrete restricted maximal function is denoted by $\mathcal{M}_{\ast,>D}^{G}$, and $\mathcal{C}(G,p)$ denotes the corresponding discrete maximal operator norm. Dimension-free estimates for q-balls. For every $p\in(1,\infty)$ and $q\in[1,\infty]$, the following two assertions are conjectured:

1. Weak form: There exist constants $C_{p,q}>0$ and $t_q>0$ such that, for every $d\in\mathbb{N}$,

$$\sup_{\|f\|_{\ell^p(\mathbb{Z}^d)}\leq 1}\big\|\mathcal{M}_{\ast,>t_qd}^{B^q(d)}f\big\|_{\ell^p(\mathbb{Z}^d)}\leq C_{p,q}.$$

2. Strong form: There exists a constant $C_{p,q}>0$ such that, for every $d\in\mathbb{N}$,

$$\mathcal{C}(B^q(d),p)\leq C_{p,q}.$$

The weak form asks for dimension-free bounds after excluding scales up to a linear threshold in $d$, while the strong form asks for dimension-free bounds for the full discrete maximal function. The preceding discussion establishes such bounds for certain ranges of $p$ and restricted scales, but the assertions above remain open for the stated ranges of $p$ and $q$.