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An Introductory Course in Lebesgue Spaces pp 383–417Cite as

Convolution and Potentials

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Abstract

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.

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References

  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. R. N. Bracewell. The Fourier transform and its applications. McGraw-Hill International Book Company, 2nd ed. edition, 1983.

    Google Scholar 

  3. H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, 2010.

    Book  Google Scholar 

  4. A.-P. Calderón. Intermediate spaces and interpolation, the complex method. Studia Math., 24:113–190, 1964.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. L. Cauchy. Cours d’Analyse de l’Ecole Royale Polytechnique: Analyse Algébrique. Debure, 1821.

    MATH  Google Scholar 

  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. J. A. Clarkson. Uniformly convex spaces. Trans. Am. Math. Soc., 40: 396–414, 1936.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. V. Cruz-Uribe and A. Fiorenza. Variable Lebesgue spaces. Foundations and harmonic analysis. New York, NY: Birkhäuser/Springer, 2013.

    Book  MATH  Google Scholar 

  10. M. Cwikel. The dual of weak L p. Ann. Inst. Fourier, 25(2):81–126, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Cwikel and C. Fefferman. Maximal seminorms on Weak L 1. Stud. Math., 69:149–154, 1980.

    MathSciNet  MATH  Google Scholar 

  12. M. Cwikel and C. Fefferman. The canonical seminorm on weak L 1. Stud. Math., 78:275–278, 1984.

    MathSciNet  MATH  Google Scholar 

  13. M. M. Day. The spaces L p with 0 < p < 1. Bull. Am. Math. Soc., 46:816–823, 1940.

    Google Scholar 

  14. M. M. Day. Reflexive Banach spaces not isomorphic to uniformly convex spaces. Bull. Am. Math. Soc., 47:313–317, 1941.

    Article  MathSciNet  MATH  Google Scholar 

  15. P. W. Day. Rearrangements of measurable functions. ProQuest LLC, Ann Arbor, MI, 1970. Thesis (Ph.D.)–California Institute of Technology.

    Google Scholar 

  16. L. Diening. Maximal function on generalized Lebesgue spaces L p(⋅ ). Math. Inequal. Appl., 7(2):245–253, 2004.

    Google Scholar 

  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. A. Fiorenza and G.E. Karadzhov. Grand and small Lebesgue spaces and their analogs. Z. Anal. Anwend., 23(4):657–681, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  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. J. García-Cuerva and J. L. Rubio de Francia. Weighted norm inequalities and related topics. North-Holland Elsevier, 1985.

    MATH  Google Scholar 

  21. G.G. Gould. On a class of integration spaces. J. Lond. Math. Soc., 34: 161–172, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  22. L. Grafakos. Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, third edition, 2014.

    Google Scholar 

  23. L. Greco, T. Iwaniec, and C. Sbordone. Inverting the p-harmonic operator. Manuscr. Math., 92(2):249–258, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. G. H. Hardy. Note on a theorem of Hilbert. Math. Z., 6:314–317, 1920.

    Article  MathSciNet  MATH  Google Scholar 

  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. G. H. Hardy and J. E. Littlewood. A maximal theorem with function-theoretic applications. Acta Math., 54(1):81–116, 1930.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  30. G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge, at the University Press, 1952. 2d ed.

    Google Scholar 

  31. L. I. Hedberg. On certain convolution inequalities. Proc. Am. Math. Soc., 36:505–510, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  32. O. Hölder. Ueber einen Mittelwertsatz. Gött. Nachr., 1889:38–47, 1889.

    MATH  Google Scholar 

  33. R.A. Hunt. An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces. Bull. Am. Math. Soc., 70:803–807, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  34. R.A. Hunt. On L(p,q) spaces. Enseign. Math. (2), 12:249–276, 1966.

    Google Scholar 

  35. T. Iwaniec and C. Sbordone. On the integrability of the Jacobian under minimal hypotheses. Arch. Ration. Mech. Anal., 119(2):129–143, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  37. J. L. W. V. Jensen. Om konvexe Funktioner og Uligheder mellem Middelvaerdier. Nyt Tidss. for Math., 16:49–68, 1905.

    Google Scholar 

  38. J. L. W. V. Jensen. Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math., 30:175–193, 1906.

    Article  MathSciNet  MATH  Google Scholar 

  39. F. Jones. Lebesgue Integration on Euclidean Space. Jones and Bartlett, 2001.

    MATH  Google Scholar 

  40. G. Köthe. Topological vector spaces I. Berlin-Heidelberg-New York: Springer Verlag 1969. XV, 456 p. (1969)., 1969.

    Google Scholar 

  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. N.S. Landkof. Foundations of modern potential theory. Springer-Verlag, 1972.

    Book  MATH  Google Scholar 

  43. G. G. Lorentz. Some new functional spaces. Ann. Math. (2), 51:37–55, 1950.

    Article  MathSciNet  MATH  Google Scholar 

  44. G.G. Lorentz. Some new functional spaces. Ann. Math. (2), 51:37–55, 1950.

    Article  MathSciNet  MATH  Google Scholar 

  45. G.G. Lorentz. On the theory of spaces Λ. Pac. J. Math., 1:411–429, 1951.

    Google Scholar 

  46. J. Lukeš and J. Malý. Measure and integral. Prague: Matfyzpress, 2nd ed. edition, 2005.

    Google Scholar 

  47. W.A.J. Luxemburg and A.C. Zaanen. Some examples of normed Köthe spaces. Math. Ann., 162:337–350, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  48. L. Maligranda. Why Hölder’s inequality should be called Rogers’ inequality. Math. Inequal. Appl., 1(1):69–83, 1998.

    MathSciNet  MATH  Google Scholar 

  49. L. Maligranda. Equivalence of the Hölder-Rogers and Minkowski inequalities. Math. Inequal. Appl., 4(2):203–207, 2001.

    MathSciNet  MATH  Google Scholar 

  50. J. Marcinkiewicz. Sur l’interpolation d’opérations. C. R. Acad. Sci., Paris, 208:1272–1273, 1939.

    MATH  Google Scholar 

  51. H. Minkowski. Geometrie der Zahlen. I. Reprint. New York: Chelsea Co., 256 p. (1953)., 1953.

    Google Scholar 

  52. D.S. Mitrinović, J.E. Pečarić, and A.M. Fink. Classical and new inequalities in analysis. Dordrecht: Kluwer Academic Publishers, 1993.

    Book  MATH  Google Scholar 

  53. B. Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc., 165:207–226, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  54. C. Niculescu and L.-E. Persson. Convex functions and their applications. A contemporary approach. New York, NY: Springer, 2006.

    Book  MATH  Google Scholar 

  55. W. Orlicz. Über konjugierte Exponentenfolgen. Stud. Math., 3: 200–211, 1931.

    MATH  Google Scholar 

  56. L. Pick, A. Kufner, O. John, and S. Fučík. Function spaces. Volume 1. 2nd revised and extended ed. Berlin: de Gruyter, 2013.

    MATH  Google Scholar 

  57. F. Riesz. Untersuchungen über Systeme integrierbarer Funktionen. Math. Ann., 69:449–497, 1910.

    Article  MathSciNet  MATH  Google Scholar 

  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. M. Riesz. Sur les maxima des formes bilinéaires et sur les fonctionelles linéaires. Acta Math., 49:465–497, 1927.

    Article  MathSciNet  MATH  Google Scholar 

  60. M. Riesz. Intégrales de Riemann-Liouville et potentiels. Acta Litt. Sci. Szeged, 9:1–42, 1938.

    MathSciNet  MATH  Google Scholar 

  61. L.J. Rogers. An extension of a certain theorem in inequalities. Messenger of mathematics, XVII(10):145–150, 1888.

    Google Scholar 

  62. W. Rudin. Real and complex analysis. New York, NY: McGraw-Hill, 1987.

    MATH  Google Scholar 

  63. J. V. Ryff. Measure preserving transformations and rearrangements. J. Math. Anal. Appl., 31:449–458, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  66. S. G. Samko. Convolution type operators in L p(x). Integral Transforms Spec. Funct., 7(1-2):123–144, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  67. I.I. Sharapudinov. Topology of the space \(\mathcal{L}^{p(t)}([0,t])\). Math. Notes, 26: 796–806, 1979.

    Google Scholar 

  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. 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.

    Article  MATH  Google Scholar 

  70. G. Sinnamon. The Fourier transform in weighted Lorentz spaces. Publ. Mat., 47(1):3–29, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  71. E. M. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, 1970.

    MATH  Google Scholar 

  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. E. M. Stein and R. Shakarchi. Fourier analysis. An Introduction. Princeton, NJ: Princeton University Press, 2003.

    MATH  Google Scholar 

  74. E. M. Stein and G. Weiss. Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, 1971.

    MATH  Google Scholar 

  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. J.D. Tamarkin and A. Zygmund. Proof of a theorem of Thorin. Bull. Am. Math. Soc., 50:279–282, 1944.

    Article  MathSciNet  MATH  Google Scholar 

  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. E. C. Titchmarsh. On conjugate functions. Proc. Lond. Math. Soc. (2), 29:49–80, 1928.

    MathSciNet  MATH  Google Scholar 

  79. E. C. Titchmarsh. Additional note on conjugate functions. J. Lond. Math. Soc., 4:204–206, 1929.

    Article  MathSciNet  MATH  Google Scholar 

  80. G. Vitali. Sui gruppi di punti e sulle funzioni di variabili reali. Torino Atti, 43:229–246, 1908.

    MATH  Google Scholar 

  81. J. von Neumann. On rings of operators. III. Ann. Math. (2), 41:94–161, 1940.

    Google Scholar 

  82. H. Weyl. Singuläre Integralgleichungen mit besonderer Berücksichtigung des Fourierschen Integraltheorems. Göttingen, 86 S (1908)., 1908.

    Google Scholar 

  83. R. L. Wheeden and A. Zygmund. Measure and integral. An introduction to real analysis. CRC Press, 2015.

    Google Scholar 

  84. N. Wiener. The ergodic theorem. Duke Math. J., 5:1–18, 1939.

    Article  MathSciNet  MATH  Google Scholar 

  85. A. Zygmund. Sur les fonctions conjuguées. C. R. Acad. Sci., Paris, 187: 1025–1026, 1928.

    MATH  Google Scholar 

  86. A. Zygmund. On a theorem of Marcinkiewicz concerning interpolation of operations. J. Math. Pures Appl. (9), 35:223–248, 1956.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Universidad Nacional de Colombia, Bogotá, Colombia

    René Erlín Castillo

  2. Pontificia Universidad Javeriana, Bogotá, Colombia

    Humberto Rafeiro

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  1. René Erlín Castillo
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Castillo, R., Rafeiro, H. (2016). Convolution and Potentials. In: An Introductory Course in Lebesgue Spaces. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-30034-4_11

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