30 problems

Let $M$ be a complete Riemannian manifold, and let $R_M$ be its Riesz transform … For $1<p<2$, write $\mathrm{L}^p(M)$ for the scalar $L^p$ space and…

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…

Reverse Riesz estimate. Then

Upper-bound conjecture for the extrapolation range. The extrapolation range is limited from above by $q_+(\mathcal{H})$.

Let $\bold{H}_{\mathbb{C}}$ denote the complex Riesz transform on $4\mathbb{R}^24$, let $k\in\mathbb{N}$, and let $1<p<\infty$. Write $q=\frac{p}{p-1}$. Carlen–Dragicevic–Kovač con…

Precise range conjecture. The precise range of exponents $p$ for which the reverse inequality is bounded should be

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…

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…

Duality conjecture. Under the framework of Hassell–Sikora, Hassell–Nardulli–Sikora, and Li, the range of $p$ for which the reverse Riesz inequality is bounded should exclude a port…

Vasilyev's conjecture. For every positive measurable function $f$ on $\mathbb R^n$, one has $\log f\in BMO(\mathbb R^n)$ if and only if there exist $c>1$, $0<\rho<1$, and a functio…

Let $G$ be a compact Lie group of dimension $n$ with a biinvariant Riemannian metric, and let $A$ be the relevant generator. Set … For $1<p<\infty$, let $\mathrm{L}^p_0(G)$ denote…

The paper studies boundedness estimates for Riesz transforms and the resulting quantitative bounds for regular Lagrangian flows on spaces with log-concave probability measures. A H…

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…

Wu's conjecture. Every element of $\mathcal{H}^1(\mathbb R^n)$ has the form $B(\omega,\gamma)$ for some $\omega,\gamma\in L^2(\mathbb R^n)$; equivalently,

Let $\Omega\subset\mathbb{R}^n$ be a smooth exterior domain, and let $\Delta_D$ and $\Delta_N$ denote the Dirichlet and Neumann Laplacians on $\Omega$, respectively. The Hassell–Si…

Covariant Riesz transform conjecture. There exists a $\kappa_0=\kappa_0(A,m)>0$, depending only on $A$ and $m$, such that for all $\kappa\geq\kappa_0$, $j\in\{1,\cdots,m\}$, and…

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

Mateu–Prat–Verdera characterization conjecture. The $s$-Riesz transform associated to $\mu$ is bounded in $L^2(\mu)$ if and only if there is a constant $\widetilde{C}\in(0,\infty)$…

Non-integer-dimensional Riesz transform conjecture. If $s\notin\mathbb{Z}$, then the $s$-Riesz transform has property $(\dagger)$.

Mateu–Prat–Verdera conjecture. The condition $$ should hold for $s1$ with $snotinmathbb{Z}$.

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 $H^1(\mathbb R^2)$ denote the real Hardy space on the plane, and let a conjugate system be a collection of singular integral transforms associated with harmonic conjugacy; the…

The non-integer dimensional Riesz transform conjecture. If $s\notin\mathbb{Z}$, then the $s$-Riesz transform has property $(\dagger)$: whenever $T_\mu$ is bounded in $L^2(\mu)$, th…

Let $1<p<\infty$, let $p^*=\max\{p,p/(p-1)\}$, and let $R^{\mathbb{R}^n}=(R_1,R_2,\ldots,R_n)$ be the vectorial Riesz transform on $\mathbb{R}^n$. Its $L^p$ operator norm is denote…