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Microlocal techniques in integral geometry. (English) Zbl 0801.44002

Integral geometry and tomography, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Arcata/CA (USA) 1989, Contemp. Math. 113, 121-135 (1991).
In a series of papers the authors have made a detailed study of some of the major problems of Gel’fand-style integral geometry (admissibility for \(X\)-ray transforms, Sobolev estimates, etc.) using microlocal techniques. This paper gives an expository overview of those papers [Duke Math. J. 58, No. 1, 205-240 (1989; Zbl 0668.44004); J. Funct. Anal. 89, No. 1, 202-232 (1990; Zbl 0717.44001); Ann. Inst. Fourier 40, No. 2, 443-466 (1990; Zbl 0726.58046)]. Recently, the authors have initiated a study of \(k\)-plane transforms for \(k > 1\) using similar techniques. Their first exposition of this work has appeared in Contemp. Math. 140, 65-71 (1992; Zbl 0791.44001)].
For the entire collection see [Zbl 0753.05015].

MSC:

44A12 Radon transform
53C65 Integral geometry
44-02 Research exposition (monographs, survey articles) pertaining to integral transforms
58J40 Pseudodifferential and Fourier integral operators on manifolds
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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