9 problems

Let $h^{\infty}$ denote the space of bounded harmonic functions on the unit ball $\mathbb{B}^n:=\{x\in\mathbb{R}^n:|x|<1\}$. For $x\in\mathbb{B}^n$ and $l\in\mathbb{S}^{n-1}$, let…

Consider the Nicole energy … and the Aratyn–Ferreira–Zimerman energy … for maps $u:\mathbb{R}^3\to\mathbb{S}^2$. A Vakulenko–Kapitansky-type bound means a lower bound of the form…

For smooth finite-energy fields $u:\mathbb{R}^3\to\mathbb{S}^2$, let … and let $Q(u)$ be the Hopf invariant. The Vakulenko–Kapitansky inequality is…

The Weinstein functional is considered on the hyperbolic space $\mathbb{H}^n$, with $H^1(\mathbb{H}^n)$ as its variational class, and the corresponding sharp value on Euclidean spa…

Let $(B^{n+1},dx^2)$ be the unit ball in Euclidean $(n+1)$-space, and let $k=\frac{n+1}{2}$. Define … For $g=e^{2u}dx^2$, let $\mathcal{F}_k(g)$ denote the critical conformal primi…

Let $g\in L^2(\mathbb{R}^4)$ be even in each variable, and let $u(x,y,z,w,t)$ solve the Schrödinger equation in $\mathbb{R}^4$ with initial data $u(x,y,z,w,0)=g(x,y,z,w)$. Four-var…

Let $g\in L^2(\mathbb{R}^3)$ be even in each variable, and let $u(x,y,z,t)$ solve the Schrödinger equation in $\mathbb{R}^3$ with initial data $u(x,y,z,0)=g(x,y,z)$. Three-variable…

Let $d\geq 3$. The quantity $\mathcal S_d$ denotes the supremum in the relevant paraboloid extension inequality over functions $\psi$ on $\mathbb{R}^d$. Foschi's Gaussian maximizer…

Let $\mathbb{H}^n$ be the quaternionic Heisenberg group, with homogeneous dimension $Q=2n+2$, homogeneous norm $|u|$ for $u=(z,t)$, and let $0<\lambda<Q$. Consider the sharp Hardy–…