On coarse embeddability into $\ell_p$-spaces and a conjecture of   Dranishnikov
  We show that the Hilbert space is coarsely embeddable into any $\ell_p$ for $1\le p<\infty$. In particular, this yields new characterizations of embeddability of separable metric spaces into the Hilbert space. 
Piotr W. Nowak