Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Suppose $f\in L^1(\mathbb{T})$ and $f$ is also differentiable. Also assume that $f'\in L^p(\mathbb{T})$ for any $1\le p\le\infty$. Suppose that the Fourier coefficients $\hat{f}(j)=0$ for all $|j|<n$. Show that $$\|f'\|_p\ge Cn\|f\|_p$$ for some constant $C$ independent of $f,p,n$.
My thought: If $f\in C^1(\mathbb{T})$, then it is easy to see that $\|f'\|_p\ge C\|f\|_p$ for some constant $C$. For the general case, should we approximate $f$ by trigonometric polynomials?
This is related to the Jackson inequalities: $$ \|f-J_n*f\|_p\leq cn^{-1}\|f'\|_p, \qquad 1\leq p<\infty, $$ where $J_n$ is the Jackson kernel. This will immediately imply your inequality for $1<p<\infty$ by boundedness of the Fourier truncation (i.e., convolution with the Dirichlet kernel). I am not sure about the case $p=1$.