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Conical square functions and non-tangential maximal functions with respect to the gaussian measure

Jan Maas, Jan van Neerven, and Pierre Portal

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We study, in $L^{1}({\mathbb R}^n;\gamma)$ with respect to the gaussian measure, non-tangential maximal functions and conical square functions associated with the Ornstein-Uhlenbeck operator by developing a set of techniques which allow us, to some extent, to compensate for the non-doubling character of the gaussian measure. The main result asserts that conical square functions can be controlled in $L^1$-norm by non-tangential maximal functions. Along the way we prove a change of aperture result for the latter. This complements recent results on gaussian Hardy spaces due to Mauceri and Meda.

Article information

Publ. Mat., Volume 55, Number 2 (2011), 313-341.

First available in Project Euclid: 22 June 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B30: $H^p$-spaces 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Hardy spaces Gaussian measure Ornstein-Uhlenbeck operator square function maximal function


Maas, Jan; van Neerven, Jan; Portal, Pierre. Conical square functions and non-tangential maximal functions with respect to the gaussian measure. Publ. Mat. 55 (2011), no. 2, 313--341.

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  • P. Auscher, A. McIntosh, and E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18(1) (2008), 192\Ndash248.
  • P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $\mathbb{R}^n$, J. Funct. Anal. 201(1) (2003), 148\Ndash184.
  • R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62(2) (1985), 304\Ndash335.
  • B. E. J. Dahlberg, C. E. Kenig, J. Pipher, and G. C. Verchota, Area integral estimates for higher order elliptic equations and systems, Ann. Inst. Fourier (Grenoble) 47(5) (1997), 1425\Ndash1461.
  • C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129(3–4) (1972), 137\Ndash193.
  • L. Forzani, R. Scotto, and W. Urbina, Riesz and Bessel potentials, the $g^k$ functions and an area function for the Gaussian measure $\gamma$, Rev. Un. Mat. Argentina 42(1) (2000), 17\Ndash37 (2001).
  • S. Hofmann and S. Mayboroda, Hardy and BMO spaces associated to divergence form elliptic operators, Math. Ann. 344(1) (2009), 37\Ndash116.
  • J. Maas, J. van Neerven, and P. Portal, Whitney coverings and the tent spaces $T^{1,q}(\gamma)$ for the Gaussian measure, Ark. Mat. (to appear); arXiv:1002.4911.
  • G. Mauceri and S. Meda, $\mathrm{BMO}$ and $H^1$ for the Ornstein\guioUhlenbeck operator, J. Funct. Anal. 252(1) (2007), 278\Ndash313.
  • G. Mauceri, S. Meda, and P. Sjögren, Endpoint estimates for first-order Riesz transforms associated to the Ornstein-Uhlenbeck operator; arXiv:1002.1240.
  • B. Muckenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc. 139 (1969), 243\Ndash260.
  • E. Pineda and W. Urbina R., Non tangential convergence for the Ornstein-Uhlenbeck semigroup, Divulg. Mat. 16(1) (2008), 107\Ndash124.
  • P. Sjögren, Operators associated with the Hermite semigroup –-asurvey, in: “Proceedings of the conference dedicated to Professor Miguel de Guzmán” (El Escorial, 1996), J. Fourier Anal. Appl. 3, Special Issue (1997), 813\Ndash823.
  • E. M. Stein, “Singular integrals and differentiability properties of functions”, Mathematical Series 30, Princeton University Press, Princeton, N.J., 1970.