Prove that $\int_U u^2\le C\int_U\int_U\frac{(u(x)-u(y))^2}{|x-y|^{n+2s}}dx\,dy$ for some $C>0$

Question

Recently I am working on some kind of fractional Sobolev inequalities, and I would like to prove that, for all $u\in W^{s,2}(U)$, $$\|u\|_{L^2(U)}\le C[u]_{W^{s,2}(U)}\qquad (\star)$$ for some $C>0$ depending only on $n,s$ and $U$, where

1. $U\subset\mathbb R^n$ is a Lipchitz domain (i.e. the boundary is sufficiently smooth);
2. $s\in(0,1)$ is a constant;
3. $$W^{s,2}(U)=\left\{f\in L^2(U):\frac{|f(x)-f(y)|}{|x-y|^{\frac n2+s}}\in L^2(U\times U)\right\}$$
4. $$\|u\|_{L^2(U)}=\sqrt{\int_U u^2}$$
5. $$[u]_{W^{s,2}(U)}=\sqrt{\int_U\int_U\frac{(u(x)-u(y))^2}{|x-y|^{n+2s}}dx\,dy}$$ is the Gagliardo seminorm.

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In particular, I need to use that to prove the first eigenvalue of the fractional Laplcian is positive. Thus, a proof not involving the positivity of the first eigenvalue is preferred.

As far as I know, the most related results is the fractional Sobolev inequality (when $p=2$), i.e. when $2s< n$, there exists $C(n,s)>0$ such that $$\|u\|_{L^{p^\star}(U)}\le C[u]_{W^{s,2}(U)}\qquad\forall u\in W^{s,2}(U)$$ where $p^\star=\frac{2n}{n-2s}$. But I cannot see how this would help to prove $(\star)$, especially when the full range of $s$ (from $0$ to $1$) is considered.

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Question: how to prove $(\star)$?