General continuous embedding in fractional Sobolev space

Question

Let $\Omega$ be a bounded domain. Then there exists a continuous embedding between the fractional Sobolev spaces $W^{s_1,p}(\Omega)\rightarrow W^{s_2,p}(\Omega)$ for $s_1>s_2$. But does there exist any embedding result from $W^{s,p}(\Omega)\rightarrow W^{s,q}(\Omega)$ for any $p>q$?

Answer 1

No. The result is not true fractional sobolev spaces. Look at this paper:

Petru Mironescu, Winfried Sickel. A Sobolev non embedding. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 2015, 26 (3), pp.291--298. ⟨hal-01162231⟩