Abstract
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
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.
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Steinerberger, S. Directional Poincaré inequalities along mixing flows. Ark Mat 54, 555–569 (2016). https://doi.org/10.1007/s11512-016-0241-7
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DOI: https://doi.org/10.1007/s11512-016-0241-7