A theorem on weak type estimates for Riesz transforms and martingale transforms

Varopoulos, Nicolas Th.

doi:10.5802/aif.826

Varopoulos, Nicolas Th.

Annales de l'Institut Fourier, Tome 31 (1981) no. 1, pp. 257-264.

Résumé
Abstract

Les transformées de Riesz d’une mesure positive singulière $ν \in M (R^{n})$ satisfont à l’inégalité faible

m [\sum_{j = 1}^{n} | R_{j} ν | > λ] \geq \frac{C ∥ ν ∥}{λ}, λ > 0

où $m$ est la mesure de Lebesgue et $C$ une constante positive dépendant de $n$ .

The Riesz transforms of a positive singular measure $ν \in M (R^{n})$ satisfy the weak type inequality

m [\sum_{j = 1}^{n} | R_{j} ν | > λ] \geq \frac{C ∥ ν ∥}{λ}, λ > 0

where $m$ denotes Lebesgue measure and $C$ is a positive constant only depending on $m$ .

MR Zbl

DOI : 10.5802/aif.826

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     author = {Varopoulos, Nicolas Th.},
     title = {A theorem on weak type estimates for {Riesz} transforms and martingale transforms},
     journal = {Annales de l'Institut Fourier},
     pages = {257--264},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {31},
     number = {1},
     year = {1981},
     doi = {10.5802/aif.826},
     mrnumber = {84e:60070},
     zbl = {0437.60003},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.826/}
}

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Varopoulos, Nicolas Th. A theorem on weak type estimates for Riesz transforms and martingale transforms. Annales de l'Institut Fourier, Tome 31 (1981) no. 1, pp. 257-264. doi : 10.5802/aif.826. http://www.numdam.org/articles/10.5802/aif.826/

Bibliographie
Cité par

[1] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University press (1970). | MR | Zbl

[2] Cereteli, Mat. Zametki, t. 22 No. 5 (1977).

[3] R.F. Gundy, On a Theorem of F. and M. Riesz and an Identity of A. Wald. (preprint). | Zbl

[4] L. Loomis, A note on the Hilbert transform, B.A.M.S., 52 (1946), 1082-1086. | MR | Zbl

[5] K. Murali Rao, Quasi-Martingales, Math. Scand., 24 (1969), 79-92. | EuDML | MR | Zbl

[6] S. Janson, Characterizations of H1 by singular integral transforms on martingales and Rn, Math. Scand., 41 (1977), 140-152. | EuDML | MR | Zbl

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