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

Author: Salem Ben Saïd
Journal: Proc. Amer. Math. Soc. 149 (2021), 279-284
MSC (2010): Primary 44A15; Secondary 39A70
DOI: https://doi.org/10.1090/proc/15206
Published electronically: October 16, 2020
MathSciNet review: 4172604
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 https://doi.org/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 https://doi.org/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 https://doi.org/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
Charles F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), no. 1, 33–60. MR 917849, DOI https://doi.org/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
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 https://doi.org/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 https://doi.org/10.1007/BF02391776
Sigurdur Helgason, The Radon transform, Progress in Mathematics, vol. 5, Birkhäuser, Boston, Mass., 1980. MR 573446
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 https://doi.org/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 https://doi.org/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
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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/0022-1236%2889%2990004-9
Sundaram Thangavelu and Yuan Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math. 97 (2005), 25–55. MR 2274972, DOI https://doi.org/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 https://doi.org/10.1080/10652460212888
Yuan Xu, Uncertainty principle on weighted spheres, balls and simplexes, J. Approx. Theory 192 (2015), 193–214. MR 3313480, DOI https://doi.org/10.1016/j.jat.2014.11.003