Consider the space
\begin{equation*}
X(\Omega) := \left\lbrace u \in L^{2}(\Omega) \iint_{\Omega\times \Omega}\frac{(u(x)-u(y))^{2}}{|x-y|^{2}}dxdy ; u= 0 \textrm{ a.e in } \mathbb{R} \setminus \Omega\right\rbrace,
\end{equation*}
where $\Omega \subset \mathbb{R}$.
We see that $X(\Omega)$ is a Hilbert space with inner product
\begin{equation*}
\left\langle u, v \right\rangle_{X(\Omega)} := \displaystyle\iint_{\Omega\times \Omega}\dfrac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{2}}dxdy.
\end{equation*}
It is well known that if $|\Omega| < \infty$ then $||u||_{L^{\infty}(\Omega)} \leq C||u||_{W^{1,2}(\Omega)}$ (see Theorem 8.8 in Brezis's Functional Analysis).
Question. Is there an analogous estimate valid for $X(\Omega)$?