Abstract
We provide a refinement of the Poincaré inequality on the torus : there exists a set of directions such that for every there is a with The derivative does not detect any oscillation in directions orthogonal to , however, for certain the geodesic flow in direction is sufficiently mixing to compensate for that defect. On the two-dimensional torus the inequality holds for but is not true for . Similar results should hold at a great level of generality on very general domains.
Citation
Stefan Steinerberger. "Directional Poincaré inequalities along mixing flows." Ark. Mat. 54 (2) 555 - 569, October 2016. https://doi.org/10.1007/s11512-016-0241-7