A Paper by Madych and Potter states that if a function $f\in W_2^k(\mathbb{R})$ has evenly spaced zeroes (i.e. if $Z(f):=\{x:f(x)=0\}$, is such that $\underset{y\in\mathbb{R}}\sup dist(y,Z(f))=h<\infty$), then we can estimate $$\|f\|_{L_2}\leq Ch^k|f|_{W_2^k}$$
My question is, is there some upper bound going the other way? That is, can we have $$|f|_{W_2^k}\leq C\|f\|_{L_2}$$ for some absolute constant, or maybe it has to be in terms of $h$.
Their bound is essentially the Poincaré inequality of order $k$: for a function vanishing on the boundary of a domain (the domain being an interval between two zeroes) we can estimate the $L^2$ norm of the function by its Sobolev norm, taking into account the size of the domain.
There is no hope for reverse inequality, because a function being small does not imply that the derivative is small. You can place zeroes wherever you want and make the graph wiggly however you want.
One situation in which it's possible to bound a derivative by the size of function itself is when the function solves an elliptic PDE.