8 problems

Let $n\ge 3$, let $\mathcal{E}$ be the class of all complex elliptic operators under study, and for $L\in\mathcal{E}$ let $p_-(L)$ be the lower endpoint of the maximal interval of…

Let $d\ge 2$, $1<p<\infty$, and let $\mathbf{R}^{(jk)}$ be the probabilistic continuous second-order Riesz transforms on $\mathbb{R}^d$. Let $p^*=\max\\{p,p/(p-1)\\}$, and let…

Coulhon–Duong conjecture. The Riesz transform is bounded on $L^p(M)$ for every $1<p<2$.

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…

Maximum principle conjecture. There is a constant $C=C(d,s)$ such that

Gaussian estimates and Riesz transform conjecture. The heat kernel of the Hodge Laplacian should have Gaussian estimates, and the Riesz transform should be bounded on $L^p$ for eve…

Let $d\geq 2$, let $0<s<d$ with $s\notin\mathbf{N}$, and let $E\subset\mathbf{R}^d$ be compact. For a gauge $\Phi$, let $\operatorname{cap}_{\Phi,s}(E)$ denote the associated nonli…

Let $M$ be a Riemannian manifold satisfying a Sobolev inequality, with non-negative Ricci curvature outside a compact set and only one end. The one-ended Riesz transform conjecture…