Let $f$ be a nice smooth function $[0,1]\to\mathbb{R}$ with $f(0)=f(1)$, $f'(0)=f'(1)$. Then it is straightforward that its Laplacian (=second derivative) is a mean-zero function, i.e., $$ \int_0^1 f''(x) dx=0. $$
Now let $L_\alpha:=(-\Delta)^{\alpha/2}$ be the fractional Laplacian (in 1 dimension). Is it true that $$ \int_0^1 L_\alpha f(x) dx=0 $$ for all $\alpha\in (0,2)$?
The Laplacian with periodic conditions would give $$ (-\Delta)^{\alpha/2}f = \sum_{n=-\infty}^{\infty}|2\pi n|^{\alpha}\hat{f}(n)e^{2\pi inx}. $$ The above appears to be mean-free because the coefficient of $e^{i0x}$ is $0$.