9 problems

Xu's conjecture. If $\lambda\leq |\alpha|_1+d-\frac{1}{2}$ and $1<p<\infty$, then $S_R^{\lambda}(L_{\alpha})f$ converges in $L^p(\mathbb{R}_{+},x^{2\alpha+1}\,dx)$ if and only if…

The square-function conjecture. For every $\alpha>\alpha(p)-\frac12$, there is a constant $C$ such that

The planar $L^{3/2}$ conjecture. The estimate

Let $-\Delta$ be the Laplacian on $\mathbb R^n$, $n\geq2$, and let $S_\lambda^\delta(-\Delta)$ be the classical Bochner–Riesz means of order $\delta$. For…

Trapezoidal sparse-bound conjecture. Sparse bounds for ${\mathcal R}_{a,t}^{\lambda}$ hold for every $\lambda>0$ and every pair $(p,q)$ satisfying

Critical-index weak-type conjecture. In this range of $p$, the operators ${\mathcal {R}}^{\lambda(p)}_{a,t}$ are of weak type $(p,p)$. This means that they map $L^p(\mathbb{R}^d)$…

Let $\psi$ be a smooth function on $I^d$, let $\chi_\circ$ be smooth and supported near the origin, and define … where $\xi=(\zeta,\tau)\in\mathbb R^{d-1}\times\mathbb R$. Assume t…

Weighted Bochner–Riesz conjecture. If

Endpoint Bochner–Riesz conjecture. For $0<\lambda\leq (n-1)/2$, $B_1^\lambda$ is of weak type $(L^{p'_\lambda},L^{p'_\lambda})$.