TAILIEUCHUNG - Đề tài " A Paley-Wiener theorem for reductive symmetric spaces "

Let X = G/H be a reductive symmetric space and K a maximal compact subgroup of G. The image under the Fourier transform of the space of K-finite compactly supported smooth functions on X is characterized. Contents 1. Introduction 2. Notation 3. The Paley-Wiener space. Main theorem 4. Pseudo wave packets 5. Generalized Eisenstein integrals 6. Induction of Arthur-Campoli relations 7. A property of the Arthur-Campoli relations 8. Proof of Theorem 9. | Annals of Mathematics A Paley- Wiener theorem for reductive symmetric spaces By E. P. van den Ban and H. Schlichtkrull Annals of Mathematics 164 2006 879 909 A Paley-Wiener theorem for reductive symmetric spaces By E. P. VAN DEN Ban and H. SCHLICHTKRULL Abstract Let X G H be a reductive symmetric space and K a maximal compact subgroup of G. The image under the Fourier transform of the space of K-finite compactly supported smooth functions on X is characterized. Contents 1. Introduction 2. Notation 3. The Paley-Wiener space. Main theorem 4. Pseudo wave packets 5. Generalized Eisenstein integrals 6. Induction of Arthur-Campoli relations 7. A property of the Arthur-Campoli relations 8. Proof of Theorem 9. A comparison of two estimates 10. A different characterization of the Paley-Wiener space 1. Introduction One of the central theorems of harmonic analysis on R is the Paley-Wiener theorem which characterizes the class of functions on C which are Fourier transforms of C -functions on R with compact support also called the Paley-Wiener-Schwartz theorem see 18 p. 249 . We consider the analogous question for the Fourier transform of a reductive symmetric space X G H that is G is a real reductive Lie group of Harish-Chandra s class and H is an open subgroup of the group Gơ of fixed points for an involution Ơ of G. The paper is a continuation of 4 and 6 in which we have shown that the Fourier transform is injective on cc X and established an inversion formula for the K-finite functions in this space with K a ơ-stable maximal compact subgroup of G. A conjectural image of the space of K-finite functions 880 E. P. VAN DEN BAN AND H. SCHLICHTKRULL in CC X was described in 4 Rem. and will be confirmed in the present paper the conjecture was already confirmed for symmetric spaces of split rank one in 4 . If G H is a Riemannian symmetric space equivalently if H is compact there is a well established theory of harmonic analysis see 17 and the Paley-Wiener theorem that we .

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