# inequalities – “Reversed” Bernstein Inequality

I’m studying harmonic analysis by myself, and I read some online notes that introduce the Bernstein inequality. One of them mention a reversed form of the Bernstein inequality, which is stated below:

Let $$mathbb{T} = mathbb{R} / mathbb{Z} = (0,1)$$ be the one-dimensional torus. Assume that a function $$f in L^{1}(mathbb{T})$$ satisfies $$hat{f}(j) = 0$$ for all $$|j| < n$$ (vanishing Fourier coefficients). Then for all $$1 leq p leq infty$$, there exists some constant $$C$$ independent of $$n,p$$ and $$f$$, such that
$$||f’||_{p} geq Cn||f||_{p}$$

It seems that an easier problem can be obtained by replacing $$f’$$ with $$f”$$ in the above inequality. The easier problem is addressed in the MO post below:

Does there exist some \$C\$ independent of \$n\$ and \$f\$ such that \$ |f”|_p geq Cn^2 | f |_p\$, where \$1 leq pleq infty\$?

However, it seems that the trick of convex Fourier coefficients used in the post above no longer applies to the harder problem (lower bounding the norm of the first derivative). Any suggestions/ideas?