# American Institute of Mathematical Sciences

2011, 18: 61-68. doi: 10.3934/era.2011.18.61

## Sharpness of Zapolsky's inequality for quasi-states and Poisson brackets

 1 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received  February 2011 Revised  May 2011 Published  July 2011

Zapolsky's inequality gives a lower bound for the $L_1$ norm of the Poisson bracket of a pair of $C^1$ functions on the two-dimensional sphere by means of quasi-states. Here we show that this lower bound is sharp.
Citation: Anat Amir. Sharpness of Zapolsky's inequality for quasi-states and Poisson brackets. Electronic Research Announcements, 2011, 18: 61-68. doi: 10.3934/era.2011.18.61
##### References:
 [1] J. F. Aarnes, Quasi-states and quasi-measures, Adv. Math., 86 (1991), 41-67. doi: 10.1016/0001-8708(91)90035-6. [2] J. F. Aarnes, Pure quasi-states and extremal quasi-measures, Math. Ann., 295 (1993), 575-588. doi: 10.1007/BF01444904. [3] J. F. Aarnes, Construction of non-sub-additive measures and discretization of Borel measures, Fund. Math., 147 (1995), 213-237. [4] A. Amir, Sharpness of Zapolsky inequality for quasi-states and Poisson brackets, preprint, arXiv:1101.1599. [5] L. Buhovsky, M. Entov and L. Polterovich, Poisson brackets and symplectic invariants, preprint, arXiv:1103.3198. [6] M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43. [7] M. Entov, L. Polterovich and F. Zapolsky, An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics, Foundations of Physics, 37 (2007), 1306-1316. doi: 10.1007/s10701-007-9158-0. [8] M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), part 1, 1037-1055. [9] V. Guillemin and A. Pollack, "Differential Topology," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1974. [10] F. F. Knudsen, Topology and the construction of extreme quasi-measures, Adv. Math., 120 (1996), 302-321. doi: 10.1006/aima.1996.0041. [11] F. F. Knudsen, New topological measures on the torus, Fund. Math., 185 (2005), 287-293. doi: 10.4064/fm185-3-6. [12] S. Lang, "Differential and Riemannian Manifolds," 3rd ed., Graduate Texts in Mathematics, 160, Springer-Verlag, New York, 1995. [13] M. E. Taylor, "Measure Theory and Integration," Graduate Studies in Mathematics, 76, American Mathematical Society, Providence, RI, 2006. [14] F. Zapolsky, Isotopy-invariant topological measures on closed orientable surfaces of higher genus, Math. Zeit., available from: http://www.springerlink.com/content/94765t4230021783/. doi: 10.1007/s00209-0100788-0. [15] F. Zapolsky, Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn., 1 (2007), 465-475. [16] F. Zapolsky, "Quasi-States and Symplectic Topology," Ph.D. thesis, Tel-Aviv University, 2009.

show all references

##### References:
 [1] J. F. Aarnes, Quasi-states and quasi-measures, Adv. Math., 86 (1991), 41-67. doi: 10.1016/0001-8708(91)90035-6. [2] J. F. Aarnes, Pure quasi-states and extremal quasi-measures, Math. Ann., 295 (1993), 575-588. doi: 10.1007/BF01444904. [3] J. F. Aarnes, Construction of non-sub-additive measures and discretization of Borel measures, Fund. Math., 147 (1995), 213-237. [4] A. Amir, Sharpness of Zapolsky inequality for quasi-states and Poisson brackets, preprint, arXiv:1101.1599. [5] L. Buhovsky, M. Entov and L. Polterovich, Poisson brackets and symplectic invariants, preprint, arXiv:1103.3198. [6] M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81 (2006), 75-99. doi: 10.4171/CMH/43. [7] M. Entov, L. Polterovich and F. Zapolsky, An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics, Foundations of Physics, 37 (2007), 1306-1316. doi: 10.1007/s10701-007-9158-0. [8] M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3 (2007), part 1, 1037-1055. [9] V. Guillemin and A. Pollack, "Differential Topology," Prentice-Hall, Inc., Englewood Cliffs, NJ, 1974. [10] F. F. Knudsen, Topology and the construction of extreme quasi-measures, Adv. Math., 120 (1996), 302-321. doi: 10.1006/aima.1996.0041. [11] F. F. Knudsen, New topological measures on the torus, Fund. Math., 185 (2005), 287-293. doi: 10.4064/fm185-3-6. [12] S. Lang, "Differential and Riemannian Manifolds," 3rd ed., Graduate Texts in Mathematics, 160, Springer-Verlag, New York, 1995. [13] M. E. Taylor, "Measure Theory and Integration," Graduate Studies in Mathematics, 76, American Mathematical Society, Providence, RI, 2006. [14] F. Zapolsky, Isotopy-invariant topological measures on closed orientable surfaces of higher genus, Math. Zeit., available from: http://www.springerlink.com/content/94765t4230021783/. doi: 10.1007/s00209-0100788-0. [15] F. Zapolsky, Quasi-states and the Poisson bracket on surfaces, J. Mod. Dyn., 1 (2007), 465-475. [16] F. Zapolsky, "Quasi-States and Symplectic Topology," Ph.D. thesis, Tel-Aviv University, 2009.
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