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.  doi: 10.1016/0001-8708(91)90035-6.  Google Scholar

[2]

J. F. Aarnes, Pure quasi-states and extremal quasi-measures,, Math. Ann., 295 (1993), 575.  doi: 10.1007/BF01444904.  Google Scholar

[3]

J. F. Aarnes, Construction of non-sub-additive measures and discretization of Borel measures,, Fund. Math., 147 (1995), 213.   Google Scholar

[4]

A. Amir, Sharpness of Zapolsky inequality for quasi-states and Poisson brackets,, preprint, ().   Google Scholar

[5]

L. Buhovsky, M. Entov and L. Polterovich, Poisson brackets and symplectic invariants,, preprint, ().   Google Scholar

[6]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections,, Comment. Math. Helv., 81 (2006), 75.  doi: 10.4171/CMH/43.  Google Scholar

[7]

M. Entov, L. Polterovich and F. Zapolsky, An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics,, Foundations of Physics, 37 (2007), 1306.  doi: 10.1007/s10701-007-9158-0.  Google Scholar

[8]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure Appl. Math. Q., 3 (2007), 1037.   Google Scholar

[9]

V. Guillemin and A. Pollack, "Differential Topology,", Prentice-Hall, (1974).   Google Scholar

[10]

F. F. Knudsen, Topology and the construction of extreme quasi-measures,, Adv. Math., 120 (1996), 302.  doi: 10.1006/aima.1996.0041.  Google Scholar

[11]

F. F. Knudsen, New topological measures on the torus,, Fund. Math., 185 (2005), 287.  doi: 10.4064/fm185-3-6.  Google Scholar

[12]

S. Lang, "Differential and Riemannian Manifolds,", 3rd ed., 160 (1995).   Google Scholar

[13]

M. E. Taylor, "Measure Theory and Integration,", Graduate Studies in Mathematics, 76 (2006).   Google Scholar

[14]

F. Zapolsky, Isotopy-invariant topological measures on closed orientable surfaces of higher genus,, Math. Zeit., ().  doi: 10.1007/s00209-0100788-0.  Google Scholar

[15]

F. Zapolsky, Quasi-states and the Poisson bracket on surfaces,, J. Mod. Dyn., 1 (2007), 465.   Google Scholar

[16]

F. Zapolsky, "Quasi-States and Symplectic Topology,", Ph.D. thesis, (2009).   Google Scholar

show all references

References:
[1]

J. F. Aarnes, Quasi-states and quasi-measures,, Adv. Math., 86 (1991), 41.  doi: 10.1016/0001-8708(91)90035-6.  Google Scholar

[2]

J. F. Aarnes, Pure quasi-states and extremal quasi-measures,, Math. Ann., 295 (1993), 575.  doi: 10.1007/BF01444904.  Google Scholar

[3]

J. F. Aarnes, Construction of non-sub-additive measures and discretization of Borel measures,, Fund. Math., 147 (1995), 213.   Google Scholar

[4]

A. Amir, Sharpness of Zapolsky inequality for quasi-states and Poisson brackets,, preprint, ().   Google Scholar

[5]

L. Buhovsky, M. Entov and L. Polterovich, Poisson brackets and symplectic invariants,, preprint, ().   Google Scholar

[6]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections,, Comment. Math. Helv., 81 (2006), 75.  doi: 10.4171/CMH/43.  Google Scholar

[7]

M. Entov, L. Polterovich and F. Zapolsky, An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics,, Foundations of Physics, 37 (2007), 1306.  doi: 10.1007/s10701-007-9158-0.  Google Scholar

[8]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure Appl. Math. Q., 3 (2007), 1037.   Google Scholar

[9]

V. Guillemin and A. Pollack, "Differential Topology,", Prentice-Hall, (1974).   Google Scholar

[10]

F. F. Knudsen, Topology and the construction of extreme quasi-measures,, Adv. Math., 120 (1996), 302.  doi: 10.1006/aima.1996.0041.  Google Scholar

[11]

F. F. Knudsen, New topological measures on the torus,, Fund. Math., 185 (2005), 287.  doi: 10.4064/fm185-3-6.  Google Scholar

[12]

S. Lang, "Differential and Riemannian Manifolds,", 3rd ed., 160 (1995).   Google Scholar

[13]

M. E. Taylor, "Measure Theory and Integration,", Graduate Studies in Mathematics, 76 (2006).   Google Scholar

[14]

F. Zapolsky, Isotopy-invariant topological measures on closed orientable surfaces of higher genus,, Math. Zeit., ().  doi: 10.1007/s00209-0100788-0.  Google Scholar

[15]

F. Zapolsky, Quasi-states and the Poisson bracket on surfaces,, J. Mod. Dyn., 1 (2007), 465.   Google Scholar

[16]

F. Zapolsky, "Quasi-States and Symplectic Topology,", Ph.D. thesis, (2009).   Google Scholar

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