Heisenberg uniqueness pairs on the Euclidean spaces and the motion group

Chattopadhyay, Arup; Ghosh, S.; Giri, D.K.; Srivastava, R.K.

Comptes Rendus. Mathématique, Tome 358 (2020) no. 3, pp. 365-377.

Dans cet article, nous considérons des paires d’unicité de Heisenberg correspondant aux courbes et surfaces exponentielles, au paraboloïde, à la sphère. De plus, nous cherchons des résultats analogues reliés à la paire d’unicité de Heisenberg sur le groupe des mouvements euclidiens et le groupe produit apparenté.

In this article, we consider Heisenberg uniqueness pairs corresponding to the exponential curve and surfaces, paraboloid, and sphere. Further, we look for analogous results related to the Heisenberg uniqueness pair on the Euclidean motion group and related product group.

Classification : 42A38, 33C10, 33C55

[1] Andrews, George E.; Askey, Richard; Roy, Ranjan Special Functions, Encyclopedia of Mathematics and Its Applications, 71, Cambridge University Press, 1999 | MR | Zbl

[2] Babot, Daniel B. Heisenberg uniqueness pairs in the plane. Three parallel lines, Proc. Am. Math. Soc., Volume 141 (2013) no. 11, pp. 3899-3904 | DOI | MR | Zbl

[3] Benedicks, Michael On Fourier transforms of functions supported on sets of finite Lebesgue measure, J. Math. Anal. Appl., Volume 106 (1985) no. 1, pp. 180-183 | DOI | MR | Zbl

[4] Canto-Martín, Francisco; Hedenmalm, Håkan; Montes-Rodríguez, Alfonso Perron–Frobenius operators and the Klein-Gordon equation, J. Eur. Math. Soc., Volume 16 (2014) no. 1, pp. 31-66 | DOI | MR | Zbl

[5] Folland, Gerald B. Real analysis. Modern techniques and their applications, Pure and Applied Mathematics, John Wiley & Sons, 1999 | Zbl

[6] Giri, Deb Kumar; Srivastava, R. K. Heisenberg uniqueness pairs for some algebraic curves in the plane, Adv. Math., Volume 310 (2017), pp. 993-1016 | DOI | MR | Zbl

[7] González Vieli, Francisco J. A uniqueness result for the Fourier transform of measures on the sphere, Bull. Aust. Math. Soc., Volume 86 (2012) no. 1, pp. 78-82 | DOI | MR | Zbl

[8] Gröchenig, Karlheinz; Jaming, Philippe The Cramér–Wold theorem on quadratic surfaces and Heisenberg uniqueness pairs, J. Inst. Math. Jussieu, Volume 19 (2020) no. 1, pp. 117-135 | DOI | Zbl

[9] Gross, Kenneth I.; Kunze, Ray A. Fourier decompositions of certain representations, Symmetric Spaces. Short courses presented at Washington University (Pure and Applied Mathematics), Volume 8, Marcel Dekker, 1972, pp. 119-139 | MR | Zbl

[10] Havin, Victor; Jöricke, Burglind The Uncertainty Principle in Harmonic Analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 28, Springer, 1994 | MR | Zbl

[11] Hedenmalm, Håkan; Montes-Rodríguez, Alfonso Heisenberg uniqueness pairs and the Klein–Gordon equation, Ann. Math., Volume 173 (2011) no. 3, pp. 1507-1527 | DOI | MR | Zbl

[12] Hedenmalm, Håkan; Montes-Rodríguez, Alfonso The Klein–Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps, J. Eur. Math. Soc. (2020) (to appear) | DOI | Zbl

[13] Hedenmalm, Håkan; Montes-Rodríguez, Alfonso The Klein–Gordon equation, the Hilbert transform, and Gauss-type maps: $H^{\infty }$ approximation, J. Anal. Math. (2020) (to appear)

[14] Heisenberg, Werner Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. f. Physik, Volume 43 (1927), pp. 172-198 | DOI | Zbl

[15] Jaming, Philippe; Kellay, Karim A dynamical system approach to Heisenberg uniqueness pairs, J. Anal. Math., Volume 134 (2018) no. 1, pp. 273-301 | DOI | MR | Zbl

[16] Kumahara, Keisaku Fourier transforms on the motion groups, J. Math. Soc. Japan, Volume 28 (1976) no. 1, pp. 18-32 | DOI | MR | Zbl

[17] Kumahara, Keisaku; Okamoto, Kiyosato An analogue of the Paley-Wiener theorem for the Euclidean motion group, Osaka J. Math., Volume 10 (1973), pp. 77-91 | MR | Zbl

[18] Lev, Nir Uniqueness theorem for Fourier transform, Bull. Sci. Math., Volume 135 (2011) no. 2, pp. 134-140 | MR | Zbl

[19] Pollicott, Mark; Yuri, Michiko Dynamical systems and ergodic theory, London Mathematical Society Student Texts, 40, London Mathematical Society, 1998 | MR | Zbl

[20] Sjölin, Per Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin, Bull. Sci. Math., Volume 135 (2011) no. 2, pp. 125-133 | DOI | MR | Zbl

[21] Sjölin, Per Heisenberg uniqueness pairs for the parabola, J. Fourier Anal. Appl., Volume 19 (2013) no. 2, pp. 410-416 | DOI | MR | Zbl

[22] Sogge, Christopher D. Oscillatory integrals and spherical harmonics, Duke Math. J., Volume 53 (1986), pp. 43-65 | DOI | MR | Zbl

[23] Srivastava, R. K. Non-harmonic cones are Heisenberg uniqueness pairs for the Fourier transform on ${ℝ}^n$, J. Fourier Anal. Appl., Volume 24 (2018) no. 6, pp. 1425-1437 | DOI | MR | Zbl

[24] Sugiura, Mitsuo Unitary representations and harmonic analysis, North-Holland Mathematical Library, 44, North-Holland, 1990 | MR | Zbl

[25] Watson, George N. A treatise on the theory of Bessel functions, Cambridge University Press, 1944 | Zbl