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Gaussian Harmonic Analysis pp 359–406Cite as

Singular Integrals with Respect to the Gaussian Measure

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Abstract

Singular integrals are among the most important operators in classical harmonic analysis.

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  • 18 December 2019

    This book was inadvertently published without updating the following corrections:

Notes

  1. 1.

    For a detailed study of this problem see, for instance, R. Weeden & A. Zygmund [294, Chapter 12], E. Stein [252, Chapter II, III], J. Duoandikoetxea [72, Chapter 4, 5], L. Grafakos [118, Chapter 4] or A. Torchinski [275, Chapter XI].

  2. 2.

    It has been mentioned before that there are several proofs of this fact (see, for instance, G. Pisier [227] or C. Gutiérrez [122])

  3. 3.

    See also [77].

References

  1. H. Aimar, L. Forzani, R. Scotto On Riesz transforms and maximal functions in the context of Gaussian harmonic analysis. Trans. Amer. Math. Soc. 359 no. 5 (2007) 2137–2154.

    Article  MathSciNet  Google Scholar 

  2. D. Bakry L’hypercontractivité et son utilisation en théorie des semigroupes. Lectures on Probability Theory, Lec. Notes in Math. 1581. Springer (1988)

    Google Scholar 

  3. Beurling, A. On two problems concerning linear operators in Hilbert space. Acta Math. 81 (1949)

    Article  MathSciNet  Google Scholar 

  4. T. Bruno Endpoint Results for the Riesz Transform of the Ornstein–Uhlenbeck Operator. Pre-print. arxiv.org/abs/1801.07214

    Google Scholar 

  5. R. R. Coifman, G. Weiss, Transference methods in analysis, Amer. Math. Soc., Providence, RI (1976) MR0481928

    Google Scholar 

  6. E. Dalmasso & R. Scotto Riesz transforms on variable Lebesgue spaces with Gaussian measure, Integral Transforms and Special Functions, 28:5, 403–420, DOI: 10.1080/10652469.2017.1296835

    Article  MathSciNet  Google Scholar 

  7. U. Dinger Weak-type (1,  1) Estimates of the Maximal Function for the Laguerre Semigroup in Finite Dimensions. Rev. Mat. Iberoam. 8 (1) (1992), 93–120.

    Google Scholar 

  8. O. Dragicevic, A. Volberg Bellman functions and dimensionless estimates of Littlewood–Paley type. J. Oper. Theory 56 (2006), 167–198. MR2261616 (2007g:42023)

    Google Scholar 

  9. J. Duoandikoetxea Fourier Analysis. Graduate Studies in Math vol 29. AMS. Providence (2001).

    Google Scholar 

  10. E. Fabes, C. Gutiérrez, R. Scotto Weak-type estimates for the Riesz transforms associated with the Gaussian measure. Rev. Mat. Iber. 10 (1994) 229–281. MR1286476 (95m:42023)

    Google Scholar 

  11. C. Fefferman, E. M. Stein H pspaces of several variables. Acta Math. 129 (1972) 137–193.

    Article  MathSciNet  Google Scholar 

  12. D. Feyel Transformations de Hilbert–Riesz gaussiennes, C. R. Acad. Sci. Paris, Sér. I Math 310 (1990), 653–655.

    Google Scholar 

  13. L. Forzani, E. Harboure, R. Scotto Weak Type Inequality for a Family of Singular Integral Operators Related with the Gaussian Measure. Potential Anal no. 31, (2009),103–116.

    Article  MathSciNet  Google Scholar 

  14. L. Forzani, R. Scotto The higher order Riesz transform for Gaussian measure need not be of weak type (1,1). Studia Math. 131 (1998), no. 3, 205–214. MR1644460 (99h:42034)

    Google Scholar 

  15. L. Forzani, R. Scotto, W. Urbina A simpler proof of theL pcontinuity of the higher order Riesz Transform with respect to the Gaussian measureγ d. Seminaire de Probabilitès XXXV Lecture Notes in Math 1755. Springer. (2001) 162–166. MR1837285 (2002c:42019)

    Google Scholar 

  16. L. Forzani, E. Sasso, R. Scotto, L pboundedness of Riesz transforms for orthogonal polynomials in a general context. Studia Math. 231 (2015), no. 1, 45–71.

    MathSciNet  MATH  Google Scholar 

  17. J. García-Cuerva, G. Mauceri, P. Sjögren, J. L. Higher-order Riesz operators for the Ornstein–Uhlenbeck semigroup. Potential Anal. 10 (1999), 379–407. MR1698617 (2000j:42024)

    Google Scholar 

  18. P. Graczyk, J. J. Loeb, I. López, A. Nowak, W. Urbina Higher order Riesz transforms, Fractional differentiation and Sobolev spaces for Laguerre expansions. J. Math. Pures Appl. (9). 84 (2005), no. 3, 375–405.

    Google Scholar 

  19. L. Grafakos Classical Fourier Analysis GTM 249-50. 2nd. Edition. Springer (2008)

    Google Scholar 

  20. R. Gundy Sur les Transformations de Riesz pour le semigroup d’Ornstein–Uhlenbeck. C.R. Acad. Sci. 303 (série I) (1986) 967–970. MR877182 (88c:60108)

    Google Scholar 

  21. C. Gutiérrez On the Riesz transforms for the Gaussian measure. J. Func. Anal. 120 (1) (1994) 107–134. MR1262249 (95c:35013)

    Google Scholar 

  22. C. Gutiérrez, A. Incognito, J. L. Torrea Riesz transforms,g-functions and multipliers for the Laguerre semigroup. Houston J. Math. 27 (2001), no. 3, 579–592.

    Google Scholar 

  23. C. Gutiérrez, C. Segovia, J. L. Torrea On higher Riesz transforms for the Gaussian measure. J. Fourier Anal. Appl. Vol 2 #6 (1996) 583–596. MR1423529 (98m:42017)

    Google Scholar 

  24. E. Harboure, R. A. Maciás, M. T. Menaŕguez, J. L. Torrea. Oscillation and variation for the Gaussian Riesz transforms and Poisson integral. Proc. R. Soc. Edinburgh Sect. A 135 (2005), no. 1, 85–104. 42B10 (44A15 47B38)

    Article  MathSciNet  Google Scholar 

  25. H. Helson, Lectures on invariant subspaces. New York. Academic Spaces (1964).

    MATH  Google Scholar 

  26. A. Korzeniowski, D. Stroock An example in the theory of hypercontractive semigroups. Proc. AMS 94 (1985), 87–90.

    Article  MathSciNet  Google Scholar 

  27. Z. K. Li Conjugated Jacobi Series and Conjugated Functions. J. Approx Theory, 86, 179–196 (1996).

    Article  MathSciNet  Google Scholar 

  28. G. Mauceri, S. Meda BMOandH 1for the Ornstein–Uhlenbeck operator. J. Funct. Anal. 252 (2007), no. 1, 278–313. MR 2357358 (2008m:42024)

    Google Scholar 

  29. G. Mauceri, S. Meda, P. Sjögren Endpoint estimates for first-order Riesz transforms associated to the Ornstein–Uhlenbeck operator., Studia Math. 224 (2014), no. 2, 153–168.

    Article  MathSciNet  Google Scholar 

  30. T. Menárguez, S. Pérez, F. Soria The Mehler maximal function: a geometric proof of weak-type 1. J. London Math. Soc. 61 (2000), no. 3, 846–856. MR1766109 (2001i:42029)

    Google Scholar 

  31. P. A. Meyer Note sur les processus d’Ornstein–Uhlenbeck. Lectures Notes in Math 920 (1982) Springer 95–133. MR770960 (86i:60150)

    Google Scholar 

  32. M. D. Morán, W. Urbina Some properties of the Gaussian Hilbert Transform. Acta Científica Venezolana, Vol 49, No 2 (1998) 102–105.

    Google Scholar 

  33. B. Muckenhoupt, Hermite conjugate expansions. Trans. Amer. Math. Soc. 139 (1969) 243–260. MR0249918 (40:3159)

    Article  MathSciNet  Google Scholar 

  34. B. Muckenhoupt, Mean Convergence of Hermite and Laguerre Series I. Trans. Amer. Math. Soc. 147 (1970) 419–431.

    Article  MathSciNet  Google Scholar 

  35. B. Muckenhoupt, E. M.Stein Classical Expansions. Trans. Amer. Math. Soc. 147 (1965) 17–92.

    Article  Google Scholar 

  36. E. Navas, W. Urbina A transference result of theL pcontinuity of the Jacobi Riesz transform to the Gaussian and Laguerre Riesz transforms. J. Fourier Anal. Appl. 19 (2013), no. 5, 910–942

    Google Scholar 

  37. F. L. Nazarov, S. Treil, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis. Algebra i Analiz 8 (1996), no. 5, 32–162 (in Russian) translation in St. Petersburg Math. J. 8 (1997), no. 5, 721–824 .

    Google Scholar 

  38. N. K. Nikol’skii Treatise on the Shift Operator: Spectral Function Theory. Springer-Verlag. Berlin (1986).

    Google Scholar 

  39. A. Nowak, On Riesz transforms for Laguerre expansions, J. Funct. Anal. 215 (2004), 217–240.

    Article  MathSciNet  Google Scholar 

  40. A. Nowak, K. Stempak L 2-Theory of Riesz Transforms for Orthogonal Expansions. J. Four. Anal. 121 (6) (2006) 675–711.

    Article  MathSciNet  Google Scholar 

  41. A. Nowak, K. Stempak A Symmetrized Conjugacy Scheme for Orthogonal Expansions. Proc. R. Soc. Edinburgh Sect. A 143 (2013), no. 2, 427–443.

    Article  MathSciNet  Google Scholar 

  42. A. Nowak, P. Sjögren Riesz Transforms for Jacobi expansions. J. Anal Math. 104 (2008) 341–369.

    Article  MathSciNet  Google Scholar 

  43. S. Pérez Estimaciones puntuales y en norma para operadores relacionados con el semigrupo de Ornstein–Uhlenbeck. Ph.D. thesis. Universidad Autónoma de Madrid (1996)

    Google Scholar 

  44. S. Pérez The local part and the strong type for operators related to the Gauss measure. J. Geom. Anal. 11 (2001), no. 3, 491–507. MR1857854 (2002h:42027)

    Article  MathSciNet  Google Scholar 

  45. S. Pérez, F. Soria Operators associated with the Ornstein–Uhlenbeck semigroup. J. London Math. Soc. 61 (2000), 857–871. MR1766110 (2001i:42030)

    Google Scholar 

  46. G. Pisier Riesz Transform: a simpler analytic proof of P. A. Meyer inequality. Séminaire de Probabilités, XXII Lectures Notes in Math 1321. Springer (1988) 485–501. MR960544 (89m:60178)

    Google Scholar 

  47. P. Portal Maximal and quadratic Gaussian Hardy spaces. 2012. Rev. Mat. Iberoam. 30 (2014), no. 1, 79–108 arXiv:1203.1998.

    Google Scholar 

  48. E. Sasso Weak Type Estimates for Riesz–Laguerre Transforms Bull. Austral. Math. Soc. Vol. 75 (2007) 397–408.

    Article  MathSciNet  Google Scholar 

  49. Sarason, D. Generalized interpolation in H ∞. Trans. Amer. Math. Soc. 127 (1967) 179–203.

    MathSciNet  MATH  Google Scholar 

  50. R. Scotto Weak type estimates for Singular Integral operators associated with the Ornstein–Uhlenbeck process. Ph.D. thesis. University of Minnesota (1993)

    Google Scholar 

  51. P. Sjögren On the maximal function for the Mehler kernel in: Mauceri G., Ricci F., Weiss G. (eds) Harmonic analysis (Cortona, 1982), Lectures Notes in Math vol 992. Springer, Berlin (1983) 73–82. MR729346 (85j:35031)

    Google Scholar 

  52. E. M. Stein Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press. Princeton (1970) .

    Google Scholar 

  53. E. M. Stein Topics in Harmonic Analysis related to the Littlewood–Paley Theory. Princeton Univ. Press. Princeton (1970). MR0252961 (40:6176)

    Google Scholar 

  54. G. Szegő Orthogonal Polynomials. Colloq. Publ. 23. Amer. Math. Soc. Providence (1959).

    Google Scholar 

  55. A. Torchinski Real variable methods in Harmonic Analysis. Acad. Press. Pure and Applied Math 123. San Diego (1986).

    Google Scholar 

  56. W. Urbina Singular Integrals with respect to the Gaussian measure. Scuola Normale Superiore di Pisa. Classe di Science. Serie IV Vol XVII, 4 (1990) 531–567. MR1093708 (92d:42010)

    Google Scholar 

  57. R. Wheeden, A. Zygmund Measure and Integral. An Introduction to Real Analysis. Marcel Dekker Inc. New York (1977)

    Google Scholar 

  58. B. Wróbel Dimension Free L p Estimates for Single Riesz Transforms via an H ∞ Joint Functional Calculus. J. Func. Anal. 267 (9) (2014) 3332–3350 MR3261112

    Google Scholar 

  59. B. Wróbel Dimension Free L p Estimates for Vectors Riesz Transforms Associated with Orthogonal Expansions. Anal PDE no 3. (2018) 745–773. MR3261112

    Google Scholar 

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  1. Mathematics and Actuarial Sciences, Roosevelt University, Chicago, IL, USA

    Wilfredo Urbina-Romero

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Urbina-Romero, W. (2019). Singular Integrals with Respect to the Gaussian Measure. In: Gaussian Harmonic Analysis. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-05597-4_9

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