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Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. For a subsetLet $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ consisting of functions $f : \mathbb{T} \to \mathbb{R}$, verifying $\int_{\mathbb{T}} f(x, y) dx dy = 0$, consider. 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 $\inf_{f \in \mathscr{C}^{\infty}_0(\mathbb{T})} \frac{\|\nabla_{irr}(f)\|_{L^2}}{\| f \|_{L^2}} > 0$?

Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. For a subset $\mathscr{C}^{\infty}_0(\mathbb{T})$ of $\mathscr{C}^{\infty}(\mathbb{T})$ consisting 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 $\inf_{f \in \mathscr{C}^{\infty}_0(\mathbb{T})} \frac{\|\nabla_{irr}(f)\|_{L^2}}{\| f \|_{L^2}} > 0$?

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 $\inf_{f \in \mathscr{C}^{\infty}_0(\mathbb{T})} \frac{\|\nabla_{irr}(f)\|_{L^2}}{\| f \|_{L^2}} > 0$?

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Sobolev-type estimate for irrational winding on a torus

Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. For a subset $\mathscr{C}^{\infty}_0(\mathbb{T})$ of $\mathscr{C}^{\infty}(\mathbb{T})$ consisting 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 $\inf_{f \in \mathscr{C}^{\infty}_0(\mathbb{T})} \frac{\|\nabla_{irr}(f)\|_{L^2}}{\| f \|_{L^2}} > 0$?