Pitt’s inequality and the uncertainty principle

Beckner, William

doi:10.1090/S0002-9939-1995-1254832-9

Pitt’s inequality and the uncertainty principle
HTML articles powered by AMS MathViewer

by William Beckner PDF

Proc. Amer. Math. Soc. 123 (1995), 1897-1905 Request permission

Abstract:

The "uncertainty principle" is formulated using logarithmic estimates obtained from a sharp form of Pitt’s inequality. The qualitative nature of this result underlies the relations connecting entropy, the Hardy-Littlewood-Sobolev inequality, and the logarithmic Sobolev inequality.

References

William Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102 (1975), no. 1, 159–182. MR 385456, DOI 10.2307/1970980
William Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 11, 4816–4819. MR 1164616, DOI 10.1073/pnas.89.11.4816
William Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer. Math. Soc. 105 (1989), no. 2, 397–400. MR 954373, DOI 10.1090/S0002-9939-1989-0954373-7
Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 420249, DOI 10.2307/2373688
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
I. I. Hirschman Jr., A note on entropy, Amer. J. Math. 79 (1957), 152–156. MR 89127, DOI 10.2307/2372390
Elliott H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. MR 717827, DOI 10.2307/2007032

Quantum mechanics

H. R. Pitt, Theorems on Fourier series and power series, Duke Math. J. 3 (1937), no. 4, 747–755. MR 1546029, DOI 10.1215/S0012-7094-37-00363-6
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Elias M. Stein, Analytic continuation of group representations, Advances in Math. 4 (1970), 172–207 (1970). MR 263985, DOI 10.1016/0001-8708(70)90022-8
E. M. Stein and Guido Weiss, Fractional integrals on $n$-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514. MR 0098285, DOI 10.1512/iumj.1958.7.57030
Fred B. Weissler, Logarithmic Sobolev inequalities for the heat-diffusion semigroup, Trans. Amer. Math. Soc. 237 (1978), 255–269. MR 479373, DOI 10.1090/S0002-9947-1978-0479373-2

The theory of groups and quantum mechanics

E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587