I found this:
\begin{equation} \lVert Wx \rVert^{2}_{L^{2}( \mathbb{R} \times \mathbb{R} )} \leq \lVert x \rVert^{2}_{L^{2}(\mathbb{R} )}, \end{equation}
which I think should be instead
\begin{equation} \lVert Wx \rVert^{2}_{L^{2}( \mathbb{R} \times \mathbb{R} )} \leq \lVert x \rVert^{2}_{L^{2}(\mathbb{R} \times \mathbb{R})}, \end{equation}
where $W$ like Wigner-Ville Distribution but Quadratic, so Quadratic nonlinear Time-Frequency Representation.
In this context
The signal $x : \mathbb{R} \rightarrow \mathbb{R}$. Assume that $x$ is continuous, $x \in C(\mathbb{R})$, $x(t) \not= 0$ only if $a \leq t \leq b$ and that $|x(t)| \leq 1$ for all $t$, so the finite signal.
Such signals span a dense subset $\mathbf D$ of $L^{2}(\mathbb{R})$, and
defining a mapping $W : \mathbf D \rightarrow L^{2}(\mathbb{R} \times \mathbb{R})$, for which the corresponding norm estimate
\begin{equation}
\lVert Wx \rVert^{2}_{L^{2}( \mathbb{R} \times \mathbb{R} )}
\leq \lVert x \rVert^{2}_{L^{2}(\mathbb{R} {\color{red} \times \mathbb{R}})},
\end{equation}
which is $L^{2}$-energy of the signal $x : \mathbb{R} \rightarrow \mathbb{C}$ in the time-frequency plane $(t,f) \in \mathbb{R} \times \mathbb{R}$.
Is my thought right that the second one is right with $L^{2}(\mathbb{R} \times \mathbb{R})$ on the right-hand-side too?