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Spherical means on two-point homogeneous spaces and applications. (English. Russian original) Zbl 1277.43015
Izv. Math. 77, No. 2, 223-252 (2013); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 2, 3-34 (2013).
This paper investigates questions of (i) injectivity, (ii) description of the kernel, and (iii) inversion of a class of operators called Pompeiu transforms. More precisely, for \(X\) a Riemannian two-point homogeneous space and \(g\) an isometry of \(X\), one may define the distribution \(g T\) associated to the compactly supported distribution \(T\) by \(\langle gT \mid f \rangle = \langle T \mid f \circ g^{-1} \rangle\). Given a family \(\mathcal{F} = \{ T_i \}_{i=1}^s\) of such distributions and an open set \(U \subset X\), if the sets \(\Lambda(U,T_i) = \{ g \in \operatorname{Isom}(X) \mid \operatorname{supp}gT_i \subset U \}\) are non-empty, one defines \(\mathcal{P}_{\mathcal{F},U}: \mathcal{C}^\infty(U) \to \times_{i=1}^s \mathcal{C}^\infty\big( \Lambda(U,T_i) \big)\) by \[ \mathcal{P}_{\mathcal{F},U} (f)(g) = \big( \langle gT_i \mid f \rangle \big)_{i=1}^s. \] This is the Pompeiu transform associated to \(\mathcal{F}\) and \(\mathcal{U}\). For example, when \(X = U = \mathbb{S}^2\) is the two-sphere and \(\mathcal{F}\) consists only in the superficial delta function of a great circle, \(\mathcal{P}_{\mathcal{F},U}\) is the Minkowski-Funk transform.
In general, (i)–(iii) are extremely difficult to solve; see [C. A. Berenstein and D. C. Struppa, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 54 , 5–111 (1989; Zbl 0706.46031)] or [L. Zalcman, NATO ASI Ser., Ser. C 365, 185–194 (1992; Zbl 0830.26005)] among many possible surveys of the problem. For example, Schiffer’s conjecture asserts that, if \(E \subset \mathbb{R}^n\) has a boundary homeomorphic to a ball, then \(\mathcal{P}_{\{ \chi_E \}, \mathbb{R}^n}\) is non-injective if and only if \(E\) is a ball.
In this paper, the authors use new methods to completely solve (i)-(iii) when \(\mathcal{F}= \{ \chi_{B_r}, \chi_{\partial B_r} \}\) and \(U = B_R\) where \(B_r\) is a ball of radius \(r\) and \(\chi_{\partial B_r}\) denotes the superficial delta function of the sphere \(\partial B_r\). Many lemmas are of independent interest and the authors give applications to problems of overdetermined interpolation in the theory of entire functions.
MSC:
43A80 Analysis on other specific Lie groups
53C65 Integral geometry
53C35 Differential geometry of symmetric spaces
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