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ArticleOriginal scientific text

Title

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

Authors 1, 2, 3

Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.
  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Canto Blanco, 28049 Madrid, Spain
  3. Current adress: Departamento de Matemáticas, Facultad de Ciencias Exactas, Universidad Nacional Salta, Bs. As. 177, 4400 Salta, Argentina

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:=d2/dx2-2xd/dx, x ∈ ℝ, need not be of weak type (1,1). A function in L1(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.

Keywords

ENG
Fourier analysis, Gaussian measure, Poisson-Hermite integrals, Hermite expansions

Bibliography

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  2. [G] Gutiérrez, C., On the Riesz transforms for the Gaussian measure, J. Funct. Anal. 120 (1994), 107-134.
  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.
  4. [M] Muckenhoupt, B., Hermite conjugate expansions, Trans. Amer. Math. Soc. 139 (1969), 243-260.
  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.
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Studia Mathematica
1998/131/3
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Pages:
205-214
Main language of publication
English
Received
1995-12-28
Published
1998
Exact and natural sciences