The higher order Riesz transform for Gaussian measure need not be of weak type (1,1)

Liliana Forzani; Roberto Scotto

Studia Mathematica (1998)

  • Volume: 131, Issue: 3, page 205-214
  • ISSN: 0039-3223

Abstract

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The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator L : = d 2 / d x 2 - 2 x d / d x , x ∈ ℝ, need not be of weak type (1,1). A function in L 1 ( d γ ) , where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.

How to cite

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Forzani, Liliana, and Scotto, Roberto. "The higher order Riesz transform for Gaussian measure need not be of weak type (1,1)." Studia Mathematica 131.3 (1998): 205-214. <http://eudml.org/doc/216576>.

@article{Forzani1998,
abstract = {The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator $L:= d^2/dx^2 - 2xd/dx$, x ∈ ℝ, need not be of weak type (1,1). A function in $L^1(dγ)$, where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.},
author = {Forzani, Liliana, Scotto, Roberto},
journal = {Studia Mathematica},
keywords = {Fourier analysis; Gaussian measure; Poisson-Hermite integrals; Hermite expansions; Riesz transform; weak-(1,1) boundedness},
language = {eng},
number = {3},
pages = {205-214},
title = {The higher order Riesz transform for Gaussian measure need not be of weak type (1,1)},
url = {http://eudml.org/doc/216576},
volume = {131},
year = {1998},
}

TY - JOUR
AU - Forzani, Liliana
AU - Scotto, Roberto
TI - The higher order Riesz transform for Gaussian measure need not be of weak type (1,1)
JO - Studia Mathematica
PY - 1998
VL - 131
IS - 3
SP - 205
EP - 214
AB - The purpose of this paper is to prove that the higher order Riesz transform for Gaussian measure associated with the Ornstein-Uhlenbeck differential operator $L:= d^2/dx^2 - 2xd/dx$, x ∈ ℝ, need not be of weak type (1,1). A function in $L^1(dγ)$, where dγ is the Gaussian measure, is given such that the distribution function of the higher order Riesz transform decays more slowly than C/λ.
LA - eng
KW - Fourier analysis; Gaussian measure; Poisson-Hermite integrals; Hermite expansions; Riesz transform; weak-(1,1) boundedness
UR - http://eudml.org/doc/216576
ER -

References

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  1. [F-G-S] Fabes, E., Gutiérrez, C. and Scotto, R., Weak-type estimates for the Riesz transforms associated with the Gaussian measure, Rev. Mat. Iberoamericana 10 (1994), 229-281. Zbl0810.42006
  2. [G] Gutiérrez, C., On the Riesz transforms for the Gaussian measure, J. Funct. Anal. 120 (1994), 107-134. Zbl0807.46030
  3. [G-S-T] Gutiérrez, C., Segovia, C. and J. L. Torrea, On higher Riesz transforms for Gaussian measures, J. Fourier Anal. Appl. 2 (1996), 583-596. Zbl0893.42007
  4. [M] Muckenhoupt, B., Hermite conjugate expansions, Trans. Amer. Math. Soc. 139 (1969), 243-260. Zbl0175.12701
  5. [Sc] Scotto, R., Weak type stimates for singular integral operators associated with the Ornstein-Uhlenbeck process, PhD thesis, University of Minnesota. 
  6. [Sj] Sjögren, P., On the maximal functions for the Mehler kernel, in: Lecture Notes in Math. 992, Springer, 1983, 73-82. 
  7. [U] Urbina, W., On singular integrals with respect to the Gaussian measure, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), 531-567. Zbl0737.42018

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