On the Sobolev and Hardy constants for the fractional Navier Laplacian
  We prove the coincidence of the Sobolev and Hardy constants relative to the "Dirichlet" and "Navier" fractional Laplacians of any real order $m\in(0,\frac{n}{2})$ over bounded domains in $\mathbb R^n$. 
Roberta Musina and Alexander I. Nazarov
math.AP