Directional Poincare inequalities along mixing flows

Abstract

We provide a refinement of the Poincar\'{e} inequality on the torus $\mathbb{T}^d$: there exists a Lebesgue-null set $\mathcal{B} \subset \mathbb{T}^d$ of directions such that for every $\alpha \in \mathcal{B}$ there is a $c_{\alpha} > 0$ with $$ \|\nabla f\|_{L^2(\mathbb{T}^d)}^{d-1} \| \left\langle \nabla f, \alpha \right\rangle\|_{L^2(\mathbb{T}^d)} \geq c_{\alpha}\|f\|_{L^2(\mathbb{T}^d)}^{d} \qquad\mbox{for all}~f\in H^1(\mathbb{T}^d)~\mbox{with mean 0.}$$ The derivative $\left\langle \nabla f, \alpha \right\rangle$ does not detect any oscillation in directions orthogonal to $\alpha$, however, for certain $\alpha$ the geodesic flow in direction $\alpha$ is sufficiently ergodic to compensate for that defect. On the two-dimensional torus $\mathbb{T}^2$ the inequality holds for $\alpha = (1, \sqrt{2})$ but fails for $\alpha = (1,e)$. Similar results should hold at a great level of generality on very general domains.