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#### Analysis of PDEsDifferential Geometrymathscidoc:1701.03017

Arkiv for Matematik, 1-15, 2015.11
We provide a refinement of the Poincaré inequality on the torus $\mathbb{T}^{d}$ : there exists a set $\mathcal{B} \subset \mathbb{T} ^{d}$ of directions such that for every $\alpha \in \mathcal{B}$ there is a $c_{\alpha } > 0$ with \begin{aligned} \|\nabla f\|_{L^{2}(\mathbb{T}^{d})}^{d-1} \| \langle \nabla f, \alpha \rangle \|_{L^{2}(\mathbb{T}^{d})} \geq c_{\alpha }\|f\| _{L^{2}(\mathbb{T}^{d})}^{d} \quad \mbox{for all}~f\in H^{1}\bigl( \mathbb{T}^{d}\bigr)~ \mbox{with mean 0.} \end{aligned} The derivative $\langle \nabla f, \alpha \rangle$ does not detect any oscillation in directions orthogonal to $\alpha$ , however, for certain $\alpha$ the geodesic flow in direction $\alpha$ is sufficiently mixing to compensate for that defect. On the two-dimensional torus $\mathbb{T}^{2}$ the inequality holds for $\alpha = (1, \sqrt{2})$ but is not true for $\alpha = (1,e)$ . Similar results should hold at a great level of generality on very general domains.
@inproceedings{stefan2015directional,
title={Directional Poincaré inequalities along mixing flows},
author={Stefan Steinerberger},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203408641188830},
booktitle={Arkiv for Matematik},
pages={1-15},
year={2015},
}

Stefan Steinerberger. Directional Poincaré inequalities along mixing flows. 2015. In Arkiv for Matematik. pp.1-15. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203408641188830.