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A support theorem for the Dunkl spherical mean operator. (English) Zbl 1468.44004

Section 1 deals with the theory of the Dunkl transform as given by C. F. Dunkl and Y. Xu [Orthogonal polynomials of several variables. 2nd ed. Cambridge: Cambridge University Press (2014; Zbl 1317.33001)], and its main properties for Euclidean spaces are discussed. The support theorem (Theorem 1.1) for the Dunkl spherical mean operator is given, where the author claims that his approach is completely different from the result previously obtained by S. Helgason [Acta Math. 113, 153–180 (1965; Zbl 0163.16602)] in the classical case. The support theorem has many applications in the field of the range theorems or for the Huygens principle. The references given in the present paper stimulate the applications and research for the support theorem.
Section 2 proves the main result (i.e., Theorem 1.1) relying on the decay of the total energy of the solutions to Euler-Poisson Darboux type equations (2.9a), (2.9b) and (2.9c). To prove the main result, the author employs Claim 2.1, Claim 2.2 and Claim 2.3, and with the help of these it is concluded that the function vanishes in the whole ball and the proof of the support theorem is completed with a different approach.

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
39A70 Difference operators
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