12 problems

For $d\geq2$, let $K_{\mathbb{H}^{(k)}}$ and $K_{R_\mathrm{dis}^{(k)}}$ be the kernels of the probabilistic and Calderón–Zygmund discrete Riesz transforms, respectively, with…

Let $R^{(k)}_\mathrm{dis}$ be the Calderón–Zygmund discrete Riesz transform on $\mathbb{Z}^d$, for $k=1,\dots,d$, and let $p^*=\max\{p,p/(p-1)\}$. The sharp norm conjecture. For al…

Let $\gamma(T)=\big(\phi_1(T),\dots,\phi_r(T)\big)\subset\mathbb{Z}[T]$ be a separated system of $r$ polynomials with integer coefficients and total degree $D_\gamma$. For a finite…

Let $\gamma(T)=\big(\phi_1(T),\dots,\phi_r(T)\big)\subset\mathbb{Z}[T]$ be a separated system of $r$ polynomials with integer coefficients and total degree $D_\gamma$. For a finite…

For integers $k\geq 3$, let $\delta_0(d,k)$ denote the exponent associated with the best known bounds for the discrete linear degree-$k$ operator, and define … This $p_0(d,k)$ prov…

Let $d\geq 1$ and let $\gamma(n)=(P_1(n),\ldots,P_d(n))$ be a polynomial curve in $\mathbb{Z}^d$, where $P_1,\ldots,P_d\in\mathbb{Z}[X]$ have separated degrees, meaning…

Optimal-exponent conjecture. The exponent $p_0(d,k)$ is conjectured to equal

Sharp-range conjecture. The linear degree-$k$ spherical maximal function is conjectured to be bounded on the sharp range $p > \frac{d}{d-k}$. If true, this would make the paper's c…

Let $\mathcal A,\mathcal B\subset\{0,\ldots,M-1\}^2$ be digit sets, let $\mathcal C_{k,\mathcal A}$ and $\mathcal C_{k,\mathcal B}$ be the associated discrete product Cantor sets,…

Generic dilation conjecture. There exists $\beta>\max(0,\frac12-\delta)$ depending only on $\delta$ such that, for a generic choice of $\alpha\in[1,M]$,

Exponential-smallness conjecture. Fix $\delta\in(1/2,1)$. There exists a sequence of pairs $(M_j,\mathcal A_j)$ such that

Gaudin-type determinantal norm conjecture. For any $\mu\in P^{\vee,+}_c$, this quadratic norm is