
## 4. Sobolev inequalities

1. #### Problem 4.1.

[P. Yang] Assuming they exist, how do the level sets of the extremals of the Sobolev inequality

$C_{N,p}\left( \int_{\mathbb{H}^n} \lvert u\rvert^{\frac{Qp}{Q-p}}\right)^{\frac{Q-p}{Q}} \leq \int_{\mathbb{H}^n} \lvert \nabla_b u\rvert^p$

vary as $p$ varies?
1. Remark. It is known that when $p=2$, the extremals are of the form

$u_\lambda = c\left( \frac{\lambda}{(1+\lambda^2\lvert z\rvert^2)^2 + \lambda^4t^2} \right)^{\frac{2}{Q-2}} .$

It is conjectured that the level sets of the extremals when $p=1$ are Pansu spheres. In particular, the conjectured level sets are different when $p=1$ and $p=2$.

Cite this as: AimPL: Analysis and geometry on pseudohermitian manifolds, available at http://aimpl.org/pshermitian.