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

top
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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.