Property about the fractional Laplacian Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \longrightarrow \mathbb{C}$ is written as
$$
f(x)=\sum_{n \in \mathbb{Z}} f_n e^{inx}
$$
then
$$
(-\Delta)^sf(x)=\sum_{n \in \mathbb{Z}} |n|^{2s} f_n e^{inx}.
$$
Question. Is true that
$$
\overline{f(x)}(-\Delta)^sf(x)=|(-\Delta)^{s/2}f(x)|^2? \tag{1}
$$
I thought in the following way: on the one-hand, we have
\begin{eqnarray}
\overline{f(x)}(-\Delta)^sf(x)= \sum_{n \in \mathbb{Z}}\overline{f_n e^{inx}}\cdot \sum_{n \in \mathbb{Z}}|n|^{2s} f_n   e^{inx}=   \sum_{n \in \mathbb{Z}} |n|^{2s} |f_n e^{inx}|^2. \tag{2}
\end{eqnarray}
On the other hand,
\begin{eqnarray}
|(-\Delta)^{s/2}f(x)|^2 &=&|(-\Delta)^{s/2}f(x)|\cdot |(-\Delta)^{s/2}f(x)| \\
&=& \sum_{n \in \mathbb{Z}} |n|^{s} |f_n e^{inx}| \cdot \sum_{n \in \mathbb{Z}} |n|^{s} |f_n e^{inx}| \\
&=& \sum_{n \in \mathbb{Z}} |n|^{2s} |f_n e^{inx}|^2. \tag{3}
\end{eqnarray}
From $(2)$ and $(3)$ follows $(1)$. That is right?
 A: No, $(2)$ is not true. However,
$$
\int_{-\pi}^{\pi} \overline{f(x)}(-\Delta)^sf(x)\; dx=  \int_{-\pi}^{\pi} |(-\Delta)^{s/2}f(x)|^2 \tag{4}
$$
is valid, for all $f \in D((-\Delta)^s)$. Indeed, by the Parseval's Identity in $[1$, page $100]$, we have
\begin{eqnarray*}
\int_{-\pi}^{\pi} |(-\Delta)^{s/2}f(x)|^2\; dx= 2\pi \sum_{n \in \mathbb{Z}} \left| (\widehat{-\Delta)^{s/2}} f (n)\right|^2 &=&  2\pi \sum_{n \in \mathbb{Z}}  \left|  |n|^{s}\widehat{f}(n) \right|^2 =2\pi \sum_{n \in \mathbb{Z}}   |n|^{2s}  \, \left|\widehat{f}(n) \right|^2 \\
&=& 2\pi \sum_{n \in \mathbb{Z}}   |n|^{2s}  \,  \widehat{f}(n)  \, \overline{\widehat{f}(n) }  \\
& = &  2\pi \sum_{n \in \mathbb{Z}}  (\widehat{-\Delta)^{s}} f (n) \, \overline{\widehat{f}(n) }  \\
& = & \int_{-\pi}^{\pi} \overline{f(x)} \, ({-\Delta)^{s}} f (x) \; dx,
\end{eqnarray*}
as claimed. The relation $(4)$, in a  sense, is a  Fractional Integration by Parts-type Identity .
$[1]$ Iorio Jr, R., Iorio, V., Fourier Analysis and Partial Differential Equations, Cambridge Studies in Advanced Mathematics $70$, Cambridge University Press, Cambridge, $2001$.
