9 problems

The $\textup{A}_\infty$ weighted Sobolev conjecture. One has

Let $M$ be a metric-measure space, let $up1$, and let $Omegasubset M$ be a connected open set containing a fixed origin $o$. Write $r(x)=\operatorname{dist}(x,o)$, and let…

Let $d=1$ and $s\in(0,\frac{1}{2})$. For $f\in\dot{H}^s(\mathbb{R})\setminus\mathcal B$, let $\mathcal Q(f)$ denote the Bianchi–Egnell quotient, let $c_{BE}(s)$ be its infimum, let…

Weak-type weighted Sobolev conjecture. For such $w$ and $f$,

Let $G$ be a graph of polynomial growth with homogeneous dimension $N\geq 3$. For the discrete $p$-biharmonic equation … where $1<p<N/2$, and for the discrete Lane–Emden system … w…

Local uniform Sobolev inequality conjecture. There exist a neighbourhood $U$ of $o$ in $N$ and a constant $C$, independent of $\delta$, such that every positive $\mathcal C^1$ func…

Uniform Sobolev inequality conjecture. The metrics $(Y,\omega_\epsilon)$ have a uniform Sobolev constant for $0<\epsilon\leqslant1$: there exists a constant $C_S$, independent of…

Sharp Sobolev trace inequality. One has

Peakedness conjecture. Within the allowable range of exponents,