Abstract
For any real numbers p,q ≥ 1, we present in this paper a (p, q)-generalized version of Beurling’s uncertainty principle for ℝn, which largely extends the classical Beurling’s theorem. We then define its analog for compact extensions of ℝn and also for Heisenberg groups.
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References
A. M. Alghamdi and A. Baklouti, “A Beurling theorem for exponential solvable Lie groups,” Journal of Lie Theory 25, 1125–1137 (2015).
S. Azaouzi, A. Baklouti, and M. Elloumi, “A generalized analog of Hardy’s uncertainty principle on compact extensions of ℝn,” Annali di Matematica 193, 723–737 (2014).
S. Azaouzi, A. Baklouti, and M. Elloumi, “Analogues of Miachy, Cowling-Price and Morgan theorem for compact extensions of ℝn,” Indian J. Pure Math. 44, 529–550 (2013).
A. Baklouti and N. Ben Salah, “On theorems of Beurlingand Cowling-Price for certain nilpotent Lie groups,” Bull. Sci. Math. 132, 529–550 (2008).
A. Baklouti, N. Ben Salah, and K. Smaoui, “Some uncertainty principles on nilpotent Lie groups,” in Contemp. Math., 363: Banach Algebras and Their Applications (Amer. Math. Soc., Providence, RI, 2004), pp. 39–52.
A. Bonami, B. Demange, and P. Jaming, “Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transform,” Revist. Math. Iberoamericana 19, 23–55 (2003).
M. G. Cowling and J. F. Price, “Generalizations of Heisenberg’s inequality,” in Lecture Notes in Math., 992: Harmonic Analysis (Springer, Berlin, 1983), pp. 443–449.
G. H. Hardy, “A theorem concerning Fourier transforms,” J. Lond. Math. Soc. 19, 227–231 (1933).
L. Hörmander, “A uniqueness theorem of Beurling for Fourier transform pairs,” Ark. Mat. 29, 237–240 (1991).
A. Kleppner and R. L. Lipsman, “The Plancherel formula for group extensions,” Ann. Sci. Éc. Norm. Super. 5(4), 459–516 (1972).
A. Kleppner and R. L. Lipsman, “The Plancherel formula for group extensions. II,” Ann. Sci. Éc. Norm. Super. 6(4), 103–132 (1973).
G. W. Mackey, The Theory of Unitary Group Representations (Chicago University Press, 1976).
R. Mneimné and F. Testard, Introduction à la théorie des de groupes de Lie classiques (Herrman, 1986).
E. K. Narayanan and S. K. Ray, “The heat kernel and Hardy’s theorem on symmetric spaces of noncompact type,” Proc. Indian Acad. Sci. Math. Sci. 112, 321–330 (2002).
R. P. Sarkar and S. Thangavelu, “On Theorems of Beurling and Hardy for the Euclidean motion group,” Tohoku Math. J., 57, 335–351 (2005).
K. Smaoui, “Beurling’s thoerem for nilpotent Lie groups,” Osaka J. Math. 48, 127–147 (2011).
S. Thangavelu, An Introduction to the Uncertainty Principle. Hardy’s Theorem on Lie Groups (Birkhäuser, 2003).
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The authors are grateful to the Deanship of Scientific Research at King Faisal University for financially supporting this work under project no. 180098.
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Dedicated to the memory of Grigorii Litvinov
The article was submitted by the authors for the English version of the journal.
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Elloumi, M., Baklouti, A. & Azaouzi, S. A Generalized Beurling Theorem for Some Lie Groups. Math Notes 107, 42–53 (2020). https://doi.org/10.1134/S0001434620010058
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DOI: https://doi.org/10.1134/S0001434620010058