Abstract
In this paper, we study the general orthogonal Radon transform \({R}_{j,k}^p\) first studied by R.S. Strichartz in [21]. The main conclusions include the sharp existence conditions for \({R}_{j,k}^pf\) on Lebesgue spaces, the relation formulas connecting our transforms with the fractional integrals and Semyanistyi integrals, through which a number of explicit inversion formulas are obtained when f restricted in the range of j-plane transforms.
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References
Fuglede, B.: An integral formula. Math. Scand. 6, 207–212 (1958)
Gonzalez, F.B.: Radon Transform on Grassmann Manifolds. Thesis, MIT, Cambridge, MA (1984)
Gonzalez, F.B.: Radon transform on Grassmann manifolds. J. Funct. Anal. 71, 339–362 (1987). https://doi.org/10.1016/0022-1236(87)90008-5
Gonzalez, F. B.: Notes on Integral Geometry and Harmonic Analysis. Lecture Note vol. 24, Math-for-Industry Lecture Note Series, Kyushu University, Faculty of Mathematics, Fukuoka (2010)
Gonzalez, F.B.: Bi-invariant differential operators on the Euclidean motion group and applications to generalized Radon transforms. Ark. Mat. 26(2), 191–204 (1988). https://doi.org/10.1007/bf02386119
Gonzalez, F.B., Kakehi, T.: Pfaffian systems and Radon transforms on affine Grassmann manifolds. Math. Ann. 326(2), 237–273 (2003). https://doi.org/10.1007/s00208-002-0398-1
Helgason, S.: Integral Geometry and Radon Transform. Springer, New York-Dordrecht-Heidelberg-London (2011)
Helgason, S.: The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Math. 113, 153–180 (1965). https://doi.org/10.1007/bf02391776
Kiryakova, V.: Generalized Fractional Calculus and Applications. Longman and J. Wiley, Harlow-N. York (1994)
Ournycheva, E., Rubin, B.: Semyanistyi’s integrals and Radon transforms on matrix spaces. J. Fourier Anal. Appl. 14(1), 60–88 (2008). https://doi.org/10.1007/s00041-007-9002-0
Rubin, B.: Reconstruction of functions from their integrals over \(k\)-planes. Israel J. of Math. 141, 93–117 (2004). https://doi.org/10.1007/BF02772213
Rubin, B.: Radon transforms on affine Grassmannians. Trans. Amer. Math. Soc. 356(12), 5045–5070 (2004). https://doi.org/10.1090/S0002-9947-04-03508-1
Rubin, B.: Introduction to Radon Transforms: With Elements of Fractional Calculus and Harmonic Analysis. Cambridge University Press, New York (2015)
Rubin, B., Wang, Y.: New inversion formulas for Radon transforms on affine Grassmannians. J. Funct. Anal. 274, 2792–2817 (2018). https://doi.org/10.1016/j.jfa.2018.03.002
Rubin, B., Wang, Y.: Riesz potentials and orthogonal Radon transforms on affine Grassmannians. Fract. Calc. Appl. Anal. 24(2), 376–392 (2021). https://doi.org/10.1515/fca-2021-0017
Rubin, B., Wang, Y.: Radon transforms for mutually orthogonal affine planes. Preprint, 2019, arXiv:1901.01150 [math.FA]
Rubin, B., Wang, Y.: Erdelyi-Kober fractional integrals and Radon transforms for mutually orthogonal affine planes. Fract. Calc. Appl. Anal. 23(4), 967–979 (2020). https://doi.org/10.1515/fca-2020-0050
Semjanistyi, V. I.: Some integral transformations and integral geometry in an elliptic space. (Russian) Trudy Sem. Vektor. Tenzor. Anal. 12, 397–441 (1963)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, New York (1993)
Stein, E.: Singular Integrals and Differentiability Properties Of Functions. Princeton Univ. Press, Princeton, NJ (1970)
Strichartz, R.S.: Harmonic analysis on Grassmannian bundles. Trans. Amer. Math. Soc. 296(1), 387–409 (1986). https://doi.org/10.1090/S0002-9947-1986-0837819-6
Acknowledgements
This work originated from the valuable discussion with Professor Boris Rubin. The author is deeply grateful to him for helpful suggestions and encouragement on this subject. I also thank the referees for their comments and valuable suggestions owing to which the original text of the paper was essentially improved. The author is supported by Guangzhou Science and Technology Bureau, under the Grant No. 202102010402.
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Wang, Y. Strichartz’s Radon transforms for mutually orthogonal affine planes and fractional integrals. Fract Calc Appl Anal 25, 1971–1993 (2022). https://doi.org/10.1007/s13540-022-00079-3
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DOI: https://doi.org/10.1007/s13540-022-00079-3