26 problems

Pérez–Rela conjecture. The inequality remains valid with

Endpoint sharp-constant conjecture. In the endpoint case $p=1$,

Let $M$ be a Riemannian manifold equipped with a Laplace–Beltrami operator, and let $P_t$ denote the associated heat semigroup. A Poincaré estimate for its heat kernel is an inequa…

Let $G=(V,E)$ be a graph with degree bound $d_*$, and let $B(x,r)$ denote the open ball of radius $r$ about $x$. Assume the degree and curvature hypotheses referred to as Assumptio…

Sharp constant conjecture. In (Arccozi), one can take

The study of Sobolev and Poincaré inequalities for differential forms in Carnot groups and, more generally, in sub-Riemannian settings remains open in full generality. Graded Lebes…

Koskela–Wildrick's conjecture. The space $\widetilde{X}$ might support some weaker Poincaré inequality, but still better than a $2$-Poincaré inequality.

Koskela–Wildrick's conjecture. The space $\widetilde{X}$ might support some weaker Poincaré inequality, but still better than a $2$-Poincaré inequality.

Let $f_\infty(|{\bf x}|)$ be an isotropic probability density on $\mathbb R^n$, and let $K(|{\bf x}|)$ be its associated weight function. Growth-rate conjecture. The function…

Let $f_\infty(|{\bf x}|)$ be an isotropic probability density on ${\mathcal I}=\{ |{\bf x}|\leq i_+\}$, with $f_\infty>0$, and define the diffusion coefficient … For a constant $c$…

Consider local Poincaré–Friedrichs inequalities on simplices, with boundary conditions imposed along some faces; in particular, compare the known improvements for triangles with th…

Let $T$ be a tetrahedron, and let $C_{{\rm PF},T,\Gamma,\ell,k,p}$ denote the Poincaré–Friedrichs constant for the relevant form degree, boundary face set, and Lebesgue exponent…

KLS conjecture. The quantity $\Psi_n$ is bounded above by a universal constant, independent of the dimension $n$.

Half-cube extremality conjecture. Indicator functions of half-cubes extremize this inequality, equivalently

Let $g(t)=\exp(t-e^t)$, and let $\gamma(x,\xi)=\exp(-a|x|-b|\xi|)$ with $a,b>0$. Suppose that $f\in L^2(\mathbb{R})$ has exponential time-frequency concentration, meaning … Let…

Let $\mu$ be a centered and isotropic probability measure on $\mathbb{R}^n$ with density $d\mu=e^{-\phi}dx$, meaning … Let $\gamma$ denote the standard Gaussian measure, and let…

Let $({\rm X},{\sf d},\mathfrak m)$ be a doubling metric measure space. For $x,y\in {\rm X}$, let $R_{x,y}^L$ denote the $L$-truncated Riesz potential with poles at $x,y$, and let…

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$. It is isotropic if it is centered and has covariance matrix equal to the identity. For a smooth function $f$, its P…

Sharp fractional Poincaré bound conjecture. The correct dependence of the bound on $\delta$ is

Let $w$ be a weight in $A_{p,\mathfrak{R}}$ for $p>1$. For a rectangle $R=I_1\times I_2$, let $\pi_R(f)$ denote the projection used in the Poincaré inequality, and let…

Let $w$ be an $A_1$ weight in $\mathbb{R}^n$, with $n\geq 2$, let $1\leq p<n$, and let $p^*$ satisfy … For a cube $Q$, write $w(Q)=\int_Qw$ and let $f_{Q,w}$ denote the weighted av…

Variance conjecture. Every log-concave isotropic random vector $X$ satisfies

Universal quadratic-bound conjecture. There exists a universal constant $C_+$ such that, for all $y$ and all $r$,

Coarse total vertical perimeter conjecture. For every measurable $A\subseteq\mathbb H(\mathbb R)$ and every $\varepsilon\in(0,1/2)$,

Let $G$ be a domain satisfying a $\beta$-quasihyperbolic boundary condition, and let $n$ be the ambient dimension. Set $p=n-n\beta$, where $p\geq 1$. Quasihyperbolic boundary condi…