9 problems

Let $D=\mathbb{R}^d\setminus\mathbb{R}^{\ell}$ with $0\leq\ell<d$, and let $\sigma_{\tau,2}=d\left(\frac{1}{\min(1,\tau)}-1\right)$. The refined localization space…

Let ${\mathcal A}$ be the parameter set for $\alpha$, let $m\in\mathbb{R}$, and suppose that the preceding proposition gives a compact embedding … for every $p\in(2,q)$. The number…

Maximal-time dichotomy conjecture.

Non-oscillation conjecture. The same conclusion holds for all $p>1$ and $q>1$.

Let $\delta_\mathcal{E}$ denote the infimum of the exponents $p$ for which the Sobolev embedding associated with the Dirichlet form $\mathcal{E}$ holds. Let $d_W\approx 2.097$ be t…

With $\mathfrak{W}$, $\Omega$, $d$, $\ell$, $k$, and $S^{d-1}$ as above, for $\alpha\in(0,d)$, the Besov--Lorentz strengthening. The Riesz potential $\operatorname{I}_{\alpha}$ map…

Let $d$ and $\ell$ be positive integers, let $S^{d-1}$ be the unit sphere in $\mathbb{R}^d$, and let $G(\ell,k)$ denote the Grassmannian of $k$-dimensional linear subspaces of…

Let $E^{2,2}_{\mathrm{rad}}(\mathbb{R}^3)$ be the radially symmetric subspace of the Coulomb–Sobolev space $E^{2,2}(\mathbb{R}^3)$. Ruiz's conjecture. … This concerns the endpoint…

Let $k,l$ be positive integers and let $\sigma\in\mathbb{R}$ be such that the set $(k,l,\sigma,\sigma)$ is non-elliptic. For $1<p<\frac43$ and $q<\infty$, consider the anisotropic…