Greenleaf, Allan; Uhlmann, Gunther 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]. Cited in 14 Documents 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 Keywords:microlocal analysis; \(X\)-ray transforms; \(k\)-plane transforms; integral geometry; Sobolev estimates Citations:Zbl 0668.44004; Zbl 0717.44001; Zbl 0726.58046; Zbl 0791.44001 PDFBibTeX XMLCite \textit{A. Greenleaf} and \textit{G. Uhlmann}, Contemp. Math. None, 121--135 (1991; Zbl 0801.44002)