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

• Autorzy: Liliana Forzani , Roberto Scotto
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Fourier analysis; Gaussian measure; Poisson-Hermite integrals; Hermite expansions

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory
• 47D03: Groups and semigroups of linear operators
• 42A99: None of the above, but in this section
• 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 131
• Numer: 3
• Strony: 205-214

Daty

wydano

1998

otrzymano

1995-12-28

Twórcy

autor

Liliana Forzani

• School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.,

autor

Roberto Scotto

• Departamento de Matemáticas, Universidad Autónoma de Madrid, Ciudad Universitaria de Canto Blanco, 28049 Madrid, Spain,
• Current adress: Departamento de Matemáticas, Facultad de Ciencias Exactas, Universidad Nacional Salta, Bs. As. 177, 4400 Salta, Argentina

Bibliografia

• [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.
• [G] Gutiérrez, C., On the Riesz transforms for the Gaussian measure, J. Funct. Anal. 120 (1994), 107-134.
• [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.
• [M] Muckenhoupt, B., Hermite conjugate expansions, Trans. Amer. Math. Soc. 139 (1969), 243-260.
• [Sc] Scotto, R., Weak type stimates for singular integral operators associated with the Ornstein-Uhlenbeck process, PhD thesis, University of Minnesota.
• [Sj] Sjögren, P., On the maximal functions for the Mehler kernel, in: Lecture Notes in Math. 992, Springer, 1983, 73-82.
• [U] Urbina, W., On singular integrals with respect to the Gaussian measure, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), 531-567.