$L^2$-Oscillation Let $f:[0,1]\to \mathbb{R}$ be a smooth function such that the following property is satisfied.
$$\int\limits_{[0,1]}\int\limits_{[0,1]}|f(x)-f(y)|^2dxdy\leq \varepsilon.$$
What can I most say about $\max\limits_{[0,1]}f-\min\limits_{[0,1]}f$?

$\def\abs#1{\left|#1\right|}$Nothing.
We have
\begin{align*}
  \int_{[0,1]^2} \abs{f(x) - f(y)}^2\, d(x,y)
   &\le 
     \int_{[0,1]^2} \abs{f(x)}^2\,d(x,y) + \int_{[0,1]^2} 2\abs{f(x)}\abs{f(y)}\, d(x,y) + \int_{[0,1]^2} \abs{f(y)}^2\, d(x,y)\\
    &= \|f\|^2 + 2\|f\|^2 + \|f\|^2\\
    &= 4\|f\|^2 
\end{align*}
That is, your term will be small if $\|f\|^2$ is small. Now let $f_n$ be a smooth, positive function with $\max f_n = n$, $\min f_n = 0$, $\mathrm{supp}\, f_n \subseteq [0, \frac 1{n^3}]$. Then $\|f\|^2 \le n^2 \cdot \frac 1{n^3} = \frac 1n$, but $\max f_n - \min f_n = n$.
