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Fractional integrals and weakly singular integral equations of the first kind in then-dimensional ball

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The purpose of the paper is to introduce and to investigate a new class of fractional integrals connected with balls in ℝn. A Riesz potentialI Ω α ρ over a ball Ω is represented by a composition of such integrals. Using this representation we obtain necessary and sufficient solvability conditions for the equationI Ω α ρ =f in the space Lpw) with a power weight w(x) and solve the equation in a closed form. The investigation is based on a special Fourier analysis adopted for operators commuting with rotations and dilations in ℝn.

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  1. [1]

    V. M. Alexandrov and E. V. Kovalenko,Problems with Mixed Boundary Conditions in Continuum Mechanics, Nauka, Moscow, 1986. (Russian)

  2. [2]

    Y. W. Chen,Entire solutions of a class of differential equations of mixed type, Comm. Pure Appl. Math.14 (1961), 229–255.

  3. [3]

    M. G. Cowling,On the Littlewood-Paley-Stein theory, Rend. Circ. Mat. Palerno29 (1981), no 1, 21–55.

  4. [4]

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

  5. [5]

    G. I. Eskin,Boundary Value Problems for Elliptic Pseudodifferential Equations, Amer. Math. Soc., Providence, R.I., 1981.

  6. [6]

    V. I. Fabrikant,Applications of Potential Theory in Mechanics: A Selection of New Results, Klumer Academic Publishers, New York, 1989.

  7. [7]

    V. I. Fabrikant,Closed form solution of a two-dimensional integral equation, Izv. Vyssh. Echebn. Zaved. Mat.2 (1971), 102–104. (Russian)

  8. [8]

    L. A. Galin (ed.),Extension of the Theory of Contact Problems in the USSR, Nauka, Moscow, 1976. (Russian)

  9. [9]

    I. C. Gohberg, and M. G. Krein,Theory and Applications of Volterra Operators in Hilbert Space, Amer. Math. Soc., Providence, R.I., 1970.

  10. [10]

    I. Gohberg and N. Krupnik,Introduction to the Theory of One-dimensional Singular Equations, Kishinev, Steentsa, 1973. German translation: Birkhäuser Verlag, Basel, 1979.

  11. [11]

    H. C. Greenwald,Lipshitz spaces of distributions on the surface of unit sphere in euclidean n-space. Pacific J. Math.70 (1970), 163–176.

  12. [12]

    M. de. Guzman,Differentiation of integrals inn, Lecture Notes in Math., Springer-Verlag, Berlin, 1975.

  13. [13]

    R. Hunt, B Muckenhoupt and R. Wheeden,Weighted norm inequalties for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc.176 (1973), 227–251.

  14. [14]

    Ch. S. Kahan,The solution of a mildly singular integral equation of the first kind on a disk, Integral Equations and Operator Theory4 (1981), 548–595.

  15. [15]

    Ch. S. Kahan,The solution of a mildly singular integral equation of the first kind on a ball, Integral Equations and Operator Theory6 (1983), 67–133.

  16. [16]

    N. K. Karapetyants and B. S. Rubin,Riesz randial potentials on the disc and fractional integration operators, Soviet Math. Sokl.25 (1082), 522–525.

  17. [17]

    N. K. Karapetyants and B. S. Rubin,Integral equations of the first kind with a weak singularity with a radial right-hand side. Differential and integral equations and complex analysis, Kalmytsk. Gos. Univ., Elista, 1986, pp. 87–106. (Russian)

  18. [18]

    P. N. Knyazev,Integral Transforms, Vyshejshaya shkola, Minsk, 1969. (Russian)

  19. [19]

    Y. L. Luke,Mathematical Functions and their Approximations, Academic Press, New York, 1975.

  20. [20]

    A. Marchaud,Sur les derivées et sur les différences des fonctions de variables reeles, J. Math. Pures Appl.6 (1927), 337–425.

  21. [21]

    B. Muckenhoupt,Weighted norm inequalities for the Hardy maximal functions, Trans. Amer. Math. Soc.165 (1972), 207–226.

  22. [22]

    S. M. Nikol’skií,Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, Berlin, 1975.

  23. [23]

    V. A. Nogin and B. S. Rubin,Inversion of parabolic potentials with L p -densities, Math. Notes39 (1986), 451–456.

  24. [24]

    V. A. Nogin and S. G. Samko,Inversion and description of Riesz potentials with densities from weighted L p -spaces, Soviet Math. (Iz. VUZ)29 (1985), 95–99.

  25. [25]

    F. W. J. Olver,Asymptotics and Special Functions, Academic Press, New York, 1974.

  26. [26]

    T. J. Osier,Open questions for research, Springer Lecture Notes in Math.457(1975), 376–381.

  27. [27]

    A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev,Integrals and Series: Elementary Functions, Nauka, Moscow 1981. (Russian)

  28. [28]

    A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichex,Integrals and Series: Supplementary Chapters, Nauka, Moscow, 1986. (Russian)

  29. [29]

    N. A. Rostovtsev,An integral equation encountered in the problem of a rigid foundation bearing on nonhomogeneous soil, J. Appl. Math. Mech.25 (1961), 238–246.

  30. [30]

    B. S. Rubin,Difference regularization of operators of potential type in L p -spaces, Math. Nachr.144 (1989), 119–147. (Russian)

  31. [31]

    B. S. Rubin,One-sided potentials, the spaces L p,r α and the inversion of Riesz and Bessel potentials in the half-space, Math. Nachr.136 (1988), 177–208.

  32. [32]

    B. S. Rubin,Fourier analysis of radial-spherical convolutions inn, Izv. Vyssh. Uchelon. Zaved. Math.I-9 (1989), 53–60;II-10 (1989), 36–44. (Russian)

  33. [33]

    B. S. Rubin,One-dimensional representation inversion and certain properties of Riesz potentials of radial functions, Math. Notes34 (1983), 751–757.

  34. [34]

    B. S. Rubin,A multidimensional integral equation of the first kind with a weak singularity on a ball, Proceedings of a Commemorative Seminar on Boundary Value problems (Minsk, 1981), “Universitetskoe”, Minsk, 1985, pp. 181–185. (Russian)

  35. [35]

    B. S. Rubin,Fractional integrals and Riesz potentials with radial density in spaces with power weight, Soviet J. Contempary Math. Anal.21(1986), 68–82.

  36. [36]

    B. S. Rubin,Inversion of potentials in ℝ n by means of Gauss-Weierstrass integrals, Math. Notes41 (1987), 22–27.

  37. [37]

    B. S. Rubin,Spaces of fractional integrals on a linear contour, Izv. Akad. Nauk Armjan. SSR, Ser. Mat.7 (1972), 373–386. (Russian)

  38. [38]

    B. S. Rubin,A method of characterization and inversion of Bessel and Riesz potentials, Soviet Math. (Iz. VUZ)30 (1986), 78–89.

  39. [39]

    S. G. Smako, A. A. Kilbas and O. I. Marichev,Integrals and derivatives of fractional order and some of their applications, “Nauka i Teknika”, Minsk, 1987. (Russian)

  40. [40]

    I. N. Sneddon,Mixed Boundary Value Problems in Potential Theory, North-Holland Publ. Co., Amsterdam, 1966.

  41. [41]

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

  42. [42]

    E. M. Stein,On limits of sequences of operators, Ann. of Math.74 (1961), 140–170.

  43. [43]

    P. Wolfe,An integral operator arising in potential theory, Applicable Anal.10 (1980), 71–80.

  44. [44]

    C. Kahane,Extension of a theorem of P. Wolfe on singular integral equations, Integral Equations and Operator Theory7 (1984), 96–117.

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Correspondence to Boris Rubin.

Additional information

Supported in part by the National 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. Fractional integrals and weakly singular integral equations of the first kind in then-dimensional ball. J. Anal. Math. 63, 55–102 (1994). https://doi.org/10.1007/BF03008419

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  • Fractional Derivative
  • Singular Integral Equation
  • Solvability Condition
  • Inversion Formula
  • Fractional Integral