We know that $$ \|u\|_{L^p(\Omega)}\leq c \|\nabla u\|_{L^p(\Omega)} $$ for $u\in W^{1,p}_0(\Omega)$. My question is there exist this inequality $$ \|u\|_{L^p(\Omega)}\leq c \Big(\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}dxdy\Big)^{1/p} $$ for $u\in W^{s,p}_0(\Omega)$, where $W^{s,p}_0(\Omega)$ is defined as the closure of $C^{\infty}_c(\Omega)$ in $(W^{s,p}(\Omega),\|\cdot\|_{W^{s,p}(\Omega)})$ under the norm
$$ \|u\|_{W^{s,p}(\Omega)}=\|u\|_{L^p(\Omega)}+\Big(\iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}dxdy\Big)^{1/p}. $$
Any help would be highly appreciated. Thanks in advance.
This is false in general. If $s \leq \frac1p$, then there is no well-defined trace operator $\mathrm{Tr} : W^{s,p}(\Omega) \to L^p(\partial\Omega)$, and as a consequence $W^{s,p}_0(\Omega) = W^{s,p}(\Omega)$. In particular, taking $u \equiv 1$ shows such an inequality cannot hold.
Provided $\Omega$ is sufficiently regular, such an inequality will hold if and only if $s > \frac1p$. The latter follows from a standard blow-up argument using a fractional analogue of Rellich-Kondrachov.
The above assertions rely on the following three properties:
If $0 < s \leq \frac1p$, then $C^{\infty}_c(\Omega)$ is dense in $W^{s,p}(\Omega)$.
If $\frac1p < s < 1$, then there is a bounded trace operator $\mathrm{Tr} : W^{s,p}(\Omega) \to L^p(\partial\Omega)$.
If $p < n$ and $q < \frac{np}{n-sp},$ then $W^{s,p}(\Omega)$ embeds compactly into $L^q(\Omega)$.
These results are somewhat difficult to track down in full, as they are scattered throughout the literature, are often stated in greater generality (usually for Besov spaces), rely on other characterisations via interpolation or Fourier analysis, and/or are only stated in the full or half space. For compact embeddings and extension theorems two self-contained references are:
Di Nezza, Eleonora; Palatucci, Giampiero; Valdinoci, Enrico, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136, No. 5, 521-573 (2012). ZBL1252.46023.
Leoni, Giovanni, A first course in Sobolev spaces, Graduate Studies in Mathematics 181. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2921-8/hbk; 978-1-4704-4226-2/ebook). xxii, 734 p. (2017). ZBL1382.46001.
Note that in Exercise 17.48 of the latter text, a Poincaré inequality valid for all $s \in (0,1)$ is stated in a ball $B$; instead of asking $u$ to vanish on the boundary, it estimates the $L^p$ norm of $u$ minus its average $(u)_B$.
Properties (1) and (2) should be contained in Chapter 2 of:
Triebel, Hans, Theory of function spaces, Monographs in Mathematics, Vol. 78. Basel-Boston-Stuttgart: Birkhäuser Verlag, DM 90.00 (1983). ZBL0546.46027./
I believe property (1) can also be verified directly by a standard cutoff argument when $s < \frac1p$, which is a instructive exercise.