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

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

Anat Amir 1,

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.
Keywords: Quasi-state..
Mathematics Subject Classification: Primary: 53D9.
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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