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The inversion of fractional integrals on a sphere

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Abstract

The purpose of the paper is to invert Riesz potentials and some other fractional integrals on then-dimensional spherical surface in ℝn+1 in the closed form. New descriptions of spaces of the fractional smoothness on a sphere are obtained in terms of spherical hypersingular integrals. It is shown that Riesz potentials of the ordersn,n + 2,n + 4, ... on a sphere are Noether operators and theird-characteristic depends on the radius of the sphere.

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References

  1. R. Askey and S. Wainger,On the behaviour of special classes of ultraspherical expansions, I, J. Anal. Math.15 (1965), 193–220.

    Article  MathSciNet  Google Scholar 

  2. H. Berens, P. L. Butzer and S. Pawelke,Limitierung verfahren von Reihen mehrdimensionaler kugelfunktionen und deren Saturationsverhalten, Publs. Res. Inst. Math. Sci. Ser. A.4 (1968), 201–268.

    MathSciNet  MATH  Google Scholar 

  3. N. Dunford and J. T. Schwartz,Linear Operators, Part I: General Theory, Interscience Publications, Inc., New York & London, 1958.

  4. A. Erdelyi (editor),Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.

    Google Scholar 

  5. I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, Academic Press, 1980.

  6. H. C. Greenwald,Lipschitz spaces on the surface of the unit sphere in Euclidean n-spaces, Pacific J. Math.50 (1974), 63–80.

    MATH  MathSciNet  Google Scholar 

  7. H. C. Greenwald,Lipschitz spaces of distributions on the surface of unit sphere in Eucledian n-space, Pacific J. Math.70 (1977), 163–176.

    MATH  MathSciNet  Google Scholar 

  8. A. Marchaud,Sur les dériviées et sur les differences des fonctions de cariables réelles, J. Math. Pures Appl.6 (1927), 337–425.

    Google Scholar 

  9. B. Muckenhoupt and E. M. Stein,Classical expansions and their relations to conjugate harmonic functions, Trans. Amer. Math. Soc.118 (1965), 17–92.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. M. Pavlov and S. G. Samko,A description of the spaces ℝ n+1 in terms of spherical hypersingular integrals, Soviet Math. Dokl.29 (1984), 549–553.

    MATH  Google Scholar 

  11. N. du Plessis,Spherical fractional integrals, Trans. Amer. Math. Soc.84 (1957), 262–272.

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Prössdorf,Some Classes of Singular Equations, North-Holland Publ. Company, Amsterdam-New York-Oxford, 1978.

    MATH  Google Scholar 

  13. B. S. Rubin,The difference regularization of operators of potential type in L p-spaces, Math. Nachr.144 (1989), 119–146 (in Russian).

    Article  MATH  MathSciNet  Google Scholar 

  14. S. G. Samko, A. A. Kilbas and O. I. Marichev,Integrals and derivatives of fractional order and some of their applications, Nauka i Tekhnika, Minsk, 1987 (in Russian).

  15. S. G. Samko,Singular integrals over a sphere and the construction of the characteristic from the symbol, Soviet Math. (Iz. VUZ)27 (1983), 35–52.

    MATH  Google Scholar 

  16. S. G. Samko,Generalized Riesz potentials and hypersingular integrals with homogenous characteristics, their symbols and inversion, Proceeding of the Steklov Inst. of Math.2 (1983), 173–243.

    Google Scholar 

  17. E. M. Stein,Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton N.J., 1970.

    MATH  Google Scholar 

  18. E. M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.

    MATH  Google Scholar 

  19. B. G. Vakulov,Theorems of Hardy-Littlewood-Sobolev type for operators of potential type in L p(Sn−1, ρ), Preprint, Rostov. Gos. Univ., Rostov-on-Don, Manuscript No. 5435-B86, deposited at VINITI (1986).

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Authors and Affiliations

  1. Department of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    Boris Rubin

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  1. Boris Rubin
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Correspondence to Boris Rubin.

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Supported in part by the National Research Council of Israel (grant no. 032-7251) and in part by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Germany).

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Rubin, B. The inversion of fractional integrals on a sphere. Israel J. Math. 79, 47–81 (1992). https://doi.org/10.1007/BF02764802

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  • DOI: https://doi.org/10.1007/BF02764802

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