Local uncertainty inequalities for locally compact groups
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- by John F. Price and Alladi Sitaram PDF
- Trans. Amer. Math. Soc. 308 (1988), 105-114 Request permission
Abstract:
Let $G$ be a locally compact unimodular group equipped with Haar measure $m$, $\hat G$ its unitary dual and $\mu$ the Plancherel measure (or something closely akin to it) on $\hat G$. When $G$ is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given $\theta \in \left [ {0,\tfrac {1} {2}} \right .)$ there exists a constant ${K_\theta }$ such that for all $f$ in a certain class of functions on $G$ and all measurable $E \subseteq \hat G$, \[ {\left ( {\int _E {\operatorname {Tr} (\pi {{(f)}^{\ast }}\pi (f)) d\mu (\pi )} } \right )^{1/2}} \leqslant {K_\theta }\mu {(E)^\theta }||{\phi _\theta }f|{|_2}\] where ${\phi _\theta }$ is a certain weight function on $G$ (for which an explicit formula is given). When $G = {{\mathbf {R}}^k}$ the inequality has been established with ${\phi _\theta }(x) = |x{|^{k\theta }}$.References
- Michael Benedicks, On Fourier transforms of functions supported on sets of finite Lebesgue measure, J. Math. Anal. Appl. 106 (1985), no. 1, 180–183. MR 780328, DOI 10.1016/0022-247X(85)90140-4
- Michael G. Cowling and John F. Price, Bandwidth versus time concentration: the Heisenberg-Pauli-Weyl inequality, SIAM J. Math. Anal. 15 (1984), no. 1, 151–165. MR 728691, DOI 10.1137/0515012
- William G. Faris, Inequalities and uncertainty principles, J. Mathematical Phys. 19 (1978), no. 2, 461–466. MR 484134, DOI 10.1063/1.523667
- Steven A. Gaal, Linear analysis and representation theory, Die Grundlehren der mathematischen Wissenschaften, Band 198, Springer-Verlag, New York-Heidelberg, 1973. MR 0447465
- Ramesh Gangolli, Spherical functions on semisimple Lie groups, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Pure and Appl. Math., Vol. 8, Dekker, New York, 1972, pp. 41–92. MR 0420157
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- I. I. Hirschman Jr., A note on entropy, Amer. J. Math. 79 (1957), 152–156. MR 89127, DOI 10.2307/2372390
- Jean Ludwig, On the Hilbert-Schmidt seminorms of $L^1$ of a nilpotent Lie group, Math. Ann. 273 (1986), no. 3, 383–395. MR 824429, DOI 10.1007/BF01450729
- John F. Price, Inequalities and local uncertainty principles, J. Math. Phys. 24 (1983), no. 7, 1711–1714. MR 709504, DOI 10.1063/1.525916 —, Sharp local uncertainty principles, Studia Math. 85 (1987), 37-45.
- John F. Price and Alladi Sitaram, Functions and their Fourier transforms with supports of finite measure for certain locally compact groups, J. Funct. Anal. 79 (1988), no. 1, 166–182. MR 950089, DOI 10.1016/0022-1236(88)90035-3
- Walter Schempp, Radar ambiguity functions, the Heisenberg group, and holomorphic theta series, Proc. Amer. Math. Soc. 92 (1984), no. 1, 103–110. MR 749901, DOI 10.1090/S0002-9939-1984-0749901-6
- Michèle Vergne, Representations of Lie groups and the orbit method, Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982) Springer, New York, 1983, pp. 59–101. MR 713793 G. Warner, Harmonic analysis on semisimple Lie groups, vols. I, II, Springer-Verlag, Berlin and New York, 1972.
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 308 (1988), 105-114
- MSC: Primary 22E30; Secondary 43A15
- DOI: https://doi.org/10.1090/S0002-9947-1988-0946433-5
- MathSciNet review: 946433