54 problems

Pérez–Rela conjecture. The inequality remains valid with

Figalli–Glaudo's nonlinear stability conjecture. For $\nu\geq 2$,

Hang–Yang conjecture. Every such solution $v$ is constant.

Let $\Omega$ and the parameters $a$, $b$, $c$, $\mu$, and $\lambda$ be as in Theorem 5.1, with $0\leqslant a<\frac{N-p}{p}$, and let the associated boundary-value problem be the eq…

Let $(M,g)$ satisfy equation … left|Rmright|{L^2(B(p0,2r))}leqepsilon0. … CS(p0,r/2)leq C V0^{-1/4}, … , $\epsilon_0$ can be chosen independently of $V_0$, and the volume lower-bou…

Let $N=2$, let $p\in(1,p_N)$, and assume that the non-degenerate bound … holds for positive constants $C_1,C_2$ and a function $\tilde g$ satisfying the stated assumptions. Let…

Concentrating-parameter criterion. It is necessary and sufficient for

Attainment conjecture. Both $\mathcal{S}_{d,2^*,\varepsilon}$ and $\tilde{\mathcal{J}}_{p,\mathcal{G}^d_\varepsilon}$ are attained.

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…

Let $n$ be the dimension of the unit sphere $athbb{S}^n$ and let $L_{2k}$ be the $k$-th GJMS operator … Let $0<l<k<n/2$. Higher-order GJMS Sobolev inequality conjecture. The follow…

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

Improved Sobolev constant conjecture. $C_S$ can be improved to

Let $N>p$ and let $\mathbb{R}^N_+=\{x\in\mathbb{R}^N:x_N>0\}$. For $p^*:=Np/(N-p)$, let $u\in C_0^\infty(\mathbb{R}^N)$ and let $\gamma>p-1$. Weighted Hardy-Sobolev-Maz'ya conjectu…

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 $S_{3/2,\mathrm{curl}}$ denote the best constant in the Sobolev-curl inequality at $p=3/2$, and let $u$ be a vector field of the form defined in the source by … . This conjectu…

Ledoux's conjecture. If this inequality holds with ${\sf C}={\sf K}_{\rm opt}$, without any curvature assumption on $(M,g)$, then

Let $\Omega$ be a bounded domain and consider the degenerate elliptic Dirichlet problem in the setting of the paper, assuming only a Sobolev inequality with no Orlicz gain. Even wh…

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain, let $1<p<\infty$, and suppose that $Q$ and $v$ satisfy $v\in L^1(\Omega)$ and $|Q|_{\mathrm{op}}^{p/2}\leq kv$. Assume that…

Maz'ya's conjecture. There is a uniform constant such that

Let the Baouendi–Grushin Sobolev inequality be the inequality denoted by, for parameters $m$, $n$, and $\alpha$, and let an extremal function mean a function attaining equality in…

Let $n\ge 2$ and $1<p<n$, and write $\mathbb{R}^n_+=\mathbb{R}^{n-1}\times\{t>0\}$. For $\varphi\in C_0^\infty(\overline{\mathbb{R}^n_+})$, define the Sobolev quotient infimum … wh…

Classification conjecture. The functions $K_{\varepsilon,\sigma_0}$ are the only nontrivial positive solutions of this equation, and the best constant in the corresponding Sobolev…

Bianchi–Egnell conjecture. For every $1<p<N$, there exists a constant $\mathcal{C}>0$ such that

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 $f$,