Convolution and Potentials

  • René Erlín Castillo
  • Humberto Rafeiro
Part of the CMS Books in Mathematics book series (CMSBM)


In this chapter we study the convolution which is a very powerful tool and some operators defined using the convolution. We first start with a detailed study about the translation operator and after that we introduce the convolution operator and give some immediate properties of the operator. As an immediate application we show that the convolution with the Gauss-Weierstrass kernel is an approximate identity operator. We also study the Young inequality for the convolution operator. The definition of a support of a convolution is given based upon the definition of the support of a (class of) function which differs from the classical definition of support of a function. Approximate identity operators are studied in a general framework via Dirac sequences and Friedrich mollifiers. We end the chapter with a succinct study of the Riesz potential.


Lebesgue Space Convolution Operator Translation Operator Approximate Identity Young Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [1]
    C. Bennett and R. Sharpley. Interpolation of operators, volume 129 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1988.Google Scholar
  2. [2]
    R. N. Bracewell. The Fourier transform and its applications. McGraw-Hill International Book Company, 2nd ed. edition, 1983.Google Scholar
  3. [3]
    H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, 2010.CrossRefGoogle Scholar
  4. [4]
    A.-P. Calderón. Intermediate spaces and interpolation, the complex method. Studia Math., 24:113–190, 1964.MathSciNetzbMATHGoogle Scholar
  5. [5]
    R. E. Castillo, F. Vallejo Narvaez, and J.C. Ramos Fernández. Multiplication and composition operators on weak l p spaces. Bull. Malays. Math. Sci. Soc., 38(3):927–973, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A. L. Cauchy. Cours d’Analyse de l’Ecole Royale Polytechnique: Analyse Algébrique. Debure, 1821.zbMATHGoogle Scholar
  7. [7]
    K. M. Chong and N. M. Rice. Equimeasurable rearrangements of functions. Queen’s University, Kingston, Ont., 1971. Queen’s Papers in Pure and Applied Mathematics, No. 28.Google Scholar
  8. [8]
    J. A. Clarkson. Uniformly convex spaces. Trans. Am. Math. Soc., 40: 396–414, 1936.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D. V. Cruz-Uribe and A. Fiorenza. Variable Lebesgue spaces. Foundations and harmonic analysis. New York, NY: Birkhäuser/Springer, 2013.CrossRefzbMATHGoogle Scholar
  10. [10]
    M. Cwikel. The dual of weak L p. Ann. Inst. Fourier, 25(2):81–126, 1975.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. Cwikel and C. Fefferman. Maximal seminorms on Weak L 1. Stud. Math., 69:149–154, 1980.MathSciNetzbMATHGoogle Scholar
  12. [12]
    M. Cwikel and C. Fefferman. The canonical seminorm on weak L 1. Stud. Math., 78:275–278, 1984.MathSciNetzbMATHGoogle Scholar
  13. [13]
    M. M. Day. The spaces L p with 0 < p < 1. Bull. Am. Math. Soc., 46:816–823, 1940.Google Scholar
  14. [14]
    M. M. Day. Reflexive Banach spaces not isomorphic to uniformly convex spaces. Bull. Am. Math. Soc., 47:313–317, 1941.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    P. W. Day. Rearrangements of measurable functions. ProQuest LLC, Ann Arbor, MI, 1970. Thesis (Ph.D.)–California Institute of Technology.Google Scholar
  16. [16]
    L. Diening. Maximal function on generalized Lebesgue spaces L p(⋅ ). Math. Inequal. Appl., 7(2):245–253, 2004.Google Scholar
  17. [17]
    L. Diening, P. Harjulehto, Hästö, and M. Růžička. Lebesgue and Sobolev spaces with variable exponents. Springer-Verlag, Lecture Notes in Mathematics, vol. 2017, Berlin, 2011.Google Scholar
  18. [18]
    A. Fiorenza and G.E. Karadzhov. Grand and small Lebesgue spaces and their analogs. Z. Anal. Anwend., 23(4):657–681, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    O. Frostman. Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddelanden Mat. Sem. Univ. Lund 3, 115 s (1935)., 1935.Google Scholar
  20. [20]
    J. García-Cuerva and J. L. Rubio de Francia. Weighted norm inequalities and related topics. North-Holland Elsevier, 1985.zbMATHGoogle Scholar
  21. [21]
    G.G. Gould. On a class of integration spaces. J. Lond. Math. Soc., 34: 161–172, 1959.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    L. Grafakos. Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, third edition, 2014.Google Scholar
  23. [23]
    L. Greco, T. Iwaniec, and C. Sbordone. Inverting the p-harmonic operator. Manuscr. Math., 92(2):249–258, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    A. Grothendieck. Réarrangements de fonctions et inégalités de convexité dans les algèbres de von Neumann munies d’une trace. In Séminaire Bourbaki, Vol. 3, pages Exp. No. 113, 127–139. Soc. Math. France, Paris, 1995.Google Scholar
  25. [25]
    J. Hadamard. Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. Journ. de Math. (4), 9:171–215, 1893.Google Scholar
  26. [26]
    G. H. Hardy. Note on a theorem of Hilbert. Math. Z., 6:314–317, 1920.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    G. H. Hardy. Note on a theorem of Hilbert concerning series of positive terms. Proc. Lond. Math. Soc. (2), 23:xlv–xlvi, 1925.Google Scholar
  28. [28]
    G. H. Hardy and J. E. Littlewood. A maximal theorem with function-theoretic applications. Acta Math., 54(1):81–116, 1930.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    G. H. Hardy, J. E. Littlewood, and G. Pólya. The maximum of a certain bilinear form. Proc. Lond. Math. Soc. (2), 25:265–282, 1926.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge, at the University Press, 1952. 2d ed.Google Scholar
  31. [31]
    L. I. Hedberg. On certain convolution inequalities. Proc. Am. Math. Soc., 36:505–510, 1973.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    O. Hölder. Ueber einen Mittelwertsatz. Gött. Nachr., 1889:38–47, 1889.zbMATHGoogle Scholar
  33. [33]
    R.A. Hunt. An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces. Bull. Am. Math. Soc., 70:803–807, 1964.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    R.A. Hunt. On L(p,q) spaces. Enseign. Math. (2), 12:249–276, 1966.Google Scholar
  35. [35]
    T. Iwaniec and C. Sbordone. On the integrability of the Jacobian under minimal hypotheses. Arch. Ration. Mech. Anal., 119(2):129–143, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    R. C. James. A non-reflexive Banach space isometric with its second conjugate space. Proc. Nat. Acad. Sci. U. S. A., 37:174–177, 1951.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    J. L. W. V. Jensen. Om konvexe Funktioner og Uligheder mellem Middelvaerdier. Nyt Tidss. for Math., 16:49–68, 1905.Google Scholar
  38. [38]
    J. L. W. V. Jensen. Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math., 30:175–193, 1906.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    F. Jones. Lebesgue Integration on Euclidean Space. Jones and Bartlett, 2001.zbMATHGoogle Scholar
  40. [40]
    G. Köthe. Topological vector spaces I. Berlin-Heidelberg-New York: Springer Verlag 1969. XV, 456 p. (1969)., 1969.Google Scholar
  41. [41]
    O. Kováčik and J. Rákosník. On spaces L p(x) and W k, p(x). Czech. Math. J., 41(4):592–618, 1991.Google Scholar
  42. [42]
    N.S. Landkof. Foundations of modern potential theory. Springer-Verlag, 1972.CrossRefzbMATHGoogle Scholar
  43. [43]
    G. G. Lorentz. Some new functional spaces. Ann. Math. (2), 51:37–55, 1950.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    G.G. Lorentz. Some new functional spaces. Ann. Math. (2), 51:37–55, 1950.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    G.G. Lorentz. On the theory of spaces Λ. Pac. J. Math., 1:411–429, 1951.Google Scholar
  46. [46]
    J. Lukeš and J. Malý. Measure and integral. Prague: Matfyzpress, 2nd ed. edition, 2005.Google Scholar
  47. [47]
    W.A.J. Luxemburg and A.C. Zaanen. Some examples of normed Köthe spaces. Math. Ann., 162:337–350, 1966.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    L. Maligranda. Why Hölder’s inequality should be called Rogers’ inequality. Math. Inequal. Appl., 1(1):69–83, 1998.MathSciNetzbMATHGoogle Scholar
  49. [49]
    L. Maligranda. Equivalence of the Hölder-Rogers and Minkowski inequalities. Math. Inequal. Appl., 4(2):203–207, 2001.MathSciNetzbMATHGoogle Scholar
  50. [50]
    J. Marcinkiewicz. Sur l’interpolation d’opérations. C. R. Acad. Sci., Paris, 208:1272–1273, 1939.zbMATHGoogle Scholar
  51. [51]
    H. Minkowski. Geometrie der Zahlen. I. Reprint. New York: Chelsea Co., 256 p. (1953)., 1953.Google Scholar
  52. [52]
    D.S. Mitrinović, J.E. Pečarić, and A.M. Fink. Classical and new inequalities in analysis. Dordrecht: Kluwer Academic Publishers, 1993.CrossRefzbMATHGoogle Scholar
  53. [53]
    B. Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc., 165:207–226, 1972.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    C. Niculescu and L.-E. Persson. Convex functions and their applications. A contemporary approach. New York, NY: Springer, 2006.CrossRefzbMATHGoogle Scholar
  55. [55]
    W. Orlicz. Über konjugierte Exponentenfolgen. Stud. Math., 3: 200–211, 1931.zbMATHGoogle Scholar
  56. [56]
    L. Pick, A. Kufner, O. John, and S. Fučík. Function spaces. Volume 1. 2nd revised and extended ed. Berlin: de Gruyter, 2013.zbMATHGoogle Scholar
  57. [57]
    F. Riesz. Untersuchungen über Systeme integrierbarer Funktionen. Math. Ann., 69:449–497, 1910.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    F. Riesz. Les systèmes d’équations linéaires à une infinite d’inconnues. Paris: Gauthier-Villars, VI + 182 S. 8. (Collection Borel.) (1913)., 1913.Google Scholar
  59. [59]
    M. Riesz. Sur les maxima des formes bilinéaires et sur les fonctionelles linéaires. Acta Math., 49:465–497, 1927.MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    M. Riesz. Intégrales de Riemann-Liouville et potentiels. Acta Litt. Sci. Szeged, 9:1–42, 1938.MathSciNetzbMATHGoogle Scholar
  61. [61]
    L.J. Rogers. An extension of a certain theorem in inequalities. Messenger of mathematics, XVII(10):145–150, 1888.Google Scholar
  62. [62]
    W. Rudin. Real and complex analysis. New York, NY: McGraw-Hill, 1987.zbMATHGoogle Scholar
  63. [63]
    J. V. Ryff. Measure preserving transformations and rearrangements. J. Math. Anal. Appl., 31:449–458, 1970.MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    H. Rafeiro and E. Rojas, Espacios de Lebesgue con exponente variable. Un espacio de Banach de funciones medibles. Caracas: Ediciones IVIC, xxii–134, 2014.Google Scholar
  65. [65]
    S. G. Samko. Convolution and potential type operators in \(L^{p(x)}(\mathbb{R}^{n})\). Integral Transforms Spec. Funct., 7(3-4):261–284, 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    S. G. Samko. Convolution type operators in L p(x). Integral Transforms Spec. Funct., 7(1-2):123–144, 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    I.I. Sharapudinov. Topology of the space \(\mathcal{L}^{p(t)}([0,t])\). Math. Notes, 26: 796–806, 1979.Google Scholar
  68. [68]
    I.I. Sharapudinov. Approximation of functions in the metric of the space L p(t) ([a,b]) and quadrature formulae. Constructive function theory, Proc. int. Conf., Varna/Bulg. 1981, 189-193 (1983)., 1983.Google Scholar
  69. [69]
    I.I. Sharapudinov. On the basis property of the Haar system in the space \(\mathcal{L}^{p(t)}([0,1])\) and the principle of localization in the mean. Math. USSR, Sb., 58: 279–287, 1987.CrossRefzbMATHGoogle Scholar
  70. [70]
    G. Sinnamon. The Fourier transform in weighted Lorentz spaces. Publ. Mat., 47(1):3–29, 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    E. M. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, 1970.zbMATHGoogle Scholar
  72. [72]
    E. M. Stein. The development of square functions in the work of A. Zygmund. Bull. Am. Math. Soc., New Ser., 7:359–376, 1982.Google Scholar
  73. [73]
    E. M. Stein and R. Shakarchi. Fourier analysis. An Introduction. Princeton, NJ: Princeton University Press, 2003.zbMATHGoogle Scholar
  74. [74]
    E. M. Stein and G. Weiss. Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, 1971.zbMATHGoogle Scholar
  75. [75]
    O. Stolz. Grundzüge der Differential- und Integralrechnung. Erster Teil: Reelle Veränderliche und Functionen. Leipzig. B. G. Teubner. X + 460 S. gr. 8 (1893)., 1893.Google Scholar
  76. [76]
    J.D. Tamarkin and A. Zygmund. Proof of a theorem of Thorin. Bull. Am. Math. Soc., 50:279–282, 1944.MathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    G.O. Thorin. An extension of a convexity theorem due to M. Riesz. Fysiogr. Sällsk. Lund Förh. 8, 166-170 (1939)., 1939.Google Scholar
  78. [78]
    E. C. Titchmarsh. On conjugate functions. Proc. Lond. Math. Soc. (2), 29:49–80, 1928.MathSciNetzbMATHGoogle Scholar
  79. [79]
    E. C. Titchmarsh. Additional note on conjugate functions. J. Lond. Math. Soc., 4:204–206, 1929.MathSciNetCrossRefzbMATHGoogle Scholar
  80. [80]
    G. Vitali. Sui gruppi di punti e sulle funzioni di variabili reali. Torino Atti, 43:229–246, 1908.zbMATHGoogle Scholar
  81. [81]
    J. von Neumann. On rings of operators. III. Ann. Math. (2), 41:94–161, 1940.Google Scholar
  82. [82]
    H. Weyl. Singuläre Integralgleichungen mit besonderer Berücksichtigung des Fourierschen Integraltheorems. Göttingen, 86 S (1908)., 1908.Google Scholar
  83. [83]
    R. L. Wheeden and A. Zygmund. Measure and integral. An introduction to real analysis. CRC Press, 2015.Google Scholar
  84. [84]
    N. Wiener. The ergodic theorem. Duke Math. J., 5:1–18, 1939.MathSciNetCrossRefzbMATHGoogle Scholar
  85. [85]
    A. Zygmund. Sur les fonctions conjuguées. C. R. Acad. Sci., Paris, 187: 1025–1026, 1928.zbMATHGoogle Scholar
  86. [86]
    A. Zygmund. On a theorem of Marcinkiewicz concerning interpolation of operations. J. Math. Pures Appl. (9), 35:223–248, 1956.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • René Erlín Castillo
    • 1
  • Humberto Rafeiro
    • 2
  1. 1.Universidad Nacional de ColombiaBogotáColombia
  2. 2.Pontificia Universidad JaverianaBogotáColombia

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