Sobolev-type estimate for irrational winding on a torus

Question

Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of functions $f : \mathbb{T} \to \mathbb{R}$, verifying $\int_{\mathbb{T}} f(x, y) dx dy = 0$. Consider a differentiation $\nabla_{irr} : \mathscr{C}^{\infty}_0(\mathbb{T}) \to \mathscr{C}^{\infty}_0(\mathbb{T})$ in an irrational direction, e.g. $\nabla_{irr}(f) = \frac{df}{dx} + \sqrt{2} \frac{df}{dy}$. Is it true that $\inf_{f \in \mathscr{C}^{\infty}_0(\mathbb{T})} \frac{\|\nabla_{irr}(f)\|_{L^2}}{\| f \|_{L^2}} > 0$?

Answer 1

No, the infimum equals zero.

In the spirit of Asaf's comment, try $$ f(x,y)=e^{i(mx+ny)} . $$ Then $\|f\|=1$, $\nabla_{irr} f = i(m+\alpha n) f$, with $\alpha=\sqrt{2}$ (but it'll work for any irrational $\alpha$), so $\|\nabla_{irr} f\|^2 = (m+\alpha n)^2$.

By Dirichlet's theorem, there are infinitely many $(m,n)$ with $|m+\alpha n| < 1/n$.