7 problems

Attainment conjecture. Both $\mathcal{S}_{d,2^*,\varepsilon}$ and $\tilde{\mathcal{J}}_{p,\mathcal{G}^d_\varepsilon}$ are attained.

Let $d\ge 2$, $1<p<\infty$, and let $\mathcal{R}^{(jk)}$ be the probabilistic discrete second-order Riesz transforms on $\mathbb{Z}^d$, $R^{(jk)}_{\mathrm{dis}}$ the standard discr…

Sign conjecture for $b_k(\alpha)$. For every $\alpha \in (0,1/3) \cup [5, \infty)$, there exists an integer $i \geq 1$ such that $b_i(\alpha)<0$; equivalently, the constants…

Let $p e(1/2,1]$ and let $f:mathbb{Z}tomathbb{R}$ be a function in $ell^{p}(mathbb{Z})$. Write ${rm Var}p f$ for the $p$-variation of $f$, and let $Mf$ denote the centered discrete…

Equality conjecture. We conjecture that $R=R_c$ when $\kappa=0$.

$S_k$ optimality conjecture. Let $C>0$. Then

Sharp variation conjecture. The constant in the inequality