A support theorem for the Dunkl spherical mean operator

Ben Saïd, Salem

doi:10.1090/proc/15206

A support theorem for the Dunkl spherical mean operator
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by Salem Ben Saïd

Proc. Amer. Math. Soc. 149 (2021), 279-284

DOI: https://doi.org/10.1090/proc/15206

Published electronically: October 16, 2020

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Abstract:

We prove a support theorem for the Dunkl spherical mean operator. The proof relies on the decay of the total energy of the solutions to Euler-Poisson-Darboux type equations.

References

Mark Agranovsky, Carlos Berenstein, and Peter Kuchment, Approximation by spherical waves in $L^p$-spaces, J. Geom. Anal. 6 (1996), no. 3, 365–383 (1997). MR 1471897, DOI 10.1007/BF02921656
Salem Ben Saïd, Sundaram Thangavelu, and Venku Naidu Dogga, Uniqueness of solutions to Schrödinger equations on $H$-type groups, J. Aust. Math. Soc. 95 (2013), no. 3, 297–314. MR 3164504, DOI 10.1017/S1446788713000311
Charles L. Epstein and Bruce Kleiner, Spherical means in annular regions, Comm. Pure Appl. Math. 46 (1993), no. 3, 441–451. MR 1202964, DOI 10.1002/cpa.3160460307
Feng Dai and Yuan Xu, Analysis on $h$-harmonics and Dunkl transforms, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser/Springer, Basel, 2015. Edited by Sergey Tikhonov. MR 3309987, DOI 10.1007/978-3-0348-0887-3
Charles F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), no. 1, 33–60. MR 917849, DOI 10.1007/BF01161629
Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 155, Cambridge University Press, Cambridge, 2014. MR 3289583, DOI 10.1017/CBO9781107786134
D. V. Gorbachev, V. I. Ivanov, and S. Yu. Tikhonov, Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications, Constr. Approx. 49 (2019), no. 3, 555–605. MR 3946421, DOI 10.1007/s00365-018-9435-5
Sigurđur Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153–180. MR 172311, DOI 10.1007/BF02391776
Sigurdur Helgason, The Radon transform, Progress in Mathematics, vol. 5, Birkhäuser, Boston, Mass., 1980. MR 573446, DOI 10.1007/978-1-4899-6765-7
S. Helgason, Some results on Radon transforms, Huygens’ principle and X-ray transforms, Integral geometry (Brunswick, Maine, 1984) Contemp. Math., vol. 63, Amer. Math. Soc., Providence, RI, 1987, pp. 151–177. MR 876318, DOI 10.1090/conm/063/876318
H. Mejjaoli and K. Trimèche, On a mean value property associated with the Dunkl Laplacian operator and applications, Integral Transform. Spec. Funct. 12 (2001), no. 3, 279–302. MR 1872437, DOI 10.1080/10652460108819351
E. K. Narayanan and S. Thangavelu, A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on $\Bbb C^n$, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 2, 459–473 (English, with English and French summaries). MR 2226023, DOI 10.5802/aif.2189
Eric Todd Quinto, Helgason’s support theorem and spherical Radon transforms, Radon transforms, geometry, and wavelets, Contemp. Math., vol. 464, Amer. Math. Soc., Providence, RI, 2008, pp. 249–264. MR 2440141, DOI 10.1090/conm/464/09088
Margit Rösler, A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355 (2003), no. 6, 2413–2438. MR 1973996, DOI 10.1090/S0002-9947-03-03235-5
Robert S. Strichartz, Harmonic analysis as spectral theory of Laplacians, J. Funct. Anal. 87 (1989), no. 1, 51–148. MR 1025883, DOI 10.1016/0022-1236(89)90004-9
Sundaram Thangavelu and Yuan Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math. 97 (2005), 25–55. MR 2274972, DOI 10.1007/BF02807401
Khalifa Trimèche, Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transforms Spec. Funct. 13 (2002), no. 1, 17–38. MR 1914125, DOI 10.1080/10652460212888
Yuan Xu, Uncertainty principle on weighted spheres, balls and simplexes, J. Approx. Theory 192 (2015), 193–214. MR 3313480, DOI 10.1016/j.jat.2014.11.003

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Proceedings of the American Mathematical Society

A support theorem for the Dunkl spherical mean operator
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Abstract: