Directional Poincare inequalities along mixing flows

TL;DR

This paper refines the Poincaré inequality on the torus by identifying directions where the inequality holds, linking ergodic properties of flows to the detection of oscillations in functions.

Contribution

It establishes directional Poincaré inequalities on the torus, connecting ergodic flow properties with the inequality's validity for specific directions.

Findings

Inequality holds for certain directions like (1,√2) on T^2.

Inequality fails for directions like (1,e) on T^2.

Ergodic flow properties influence the inequality's applicability.

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.