Let's consider a smooth function $f:\mathbb{R}\to \mathbb{R}$ and the quantity $$ q(f):= \frac{|f(0)|}{\lVert f\rVert_{L^1(-1,1)}}. $$ If $q(f)$ is rather large it means that $f$ is relatively small at most points in the interval $(-1,1)$ but not at the midpoint $0$. In this case $f$ must be somewhat spiky at the center. Intuitively, $q(f)$ may then be understood as a measure for (non)-smoothness.
I would be interested in whether there is a more formal statement to this intuition. For instance, is there a natural and large class of smooth functions $\mathcal{F}\subseteq L^2(\mathbb{R})$ with the property that $q(f)$ is bounded on $\mathcal{F}$?
