6 problems

An H-type group is a two-step nilpotent Lie group equipped with an inner product for which the associated Kaplan maps satisfy the H-type condition. Frank and Lieb's Hardy–Littlewoo…

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 $u,v>0$ solve the critical Hardy–Littlewood–Sobolev system … or the corresponding fractional PDE system … in $\mathbb{R}^n$, where $0<p,q<\infty$ and…

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–…

Let $\alpha\in(0,n)$, let $p,q$ be positive exponents, and consider the Hardy–Littlewood–Sobolev system … Define the critical Sobolev hyperbola by…

Subcritical nonexistence conjecture. The system has no positive solution whenever