Let $\Omega\subset\mathbb{H}$ is bounded domain where $\mathbb{H}$ is the Heisenberg group with homogeneous dimension $2N+2$. Then the Sobolev space $W^{1,p}(\Omega)$ is defined as the space of all measurable functions $u:\Omega\to\mathbb{R}$ such that $u,\nabla_{H}u\in L^p(\Omega)$ under the norm $$ ||u||=(||u||_{p}+||\nabla_{H}u||_{p}), $$ where $||\cdot||_p$ denots the $L^p(\Omega)$ norm.
Then (i) Is $W^{1,p}(\Omega)$ uniformly convex for $p>1$? (ii) Is the mapping $W^{1,p}(\Omega)\to L^p(\Omega)$ compact?
I do not get any counterexample neither some reference of the proof.
Can somebody kindly help me?
Thanks in advance.