A theorem on weak type estimates for Riesz transforms and martingale transforms
Annales de l'Institut Fourier, Volume 31 (1981) no. 1, p. 257-264

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

$m\left[\sum _{j=1}^{n}|{R}_{j}\nu |>\lambda \right]\ge \frac{C\parallel \nu \parallel }{\lambda },\phantom{\rule{3.33333pt}{0ex}}\lambda >0$

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

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

$m\left[\sum _{j=1}^{n}|{R}_{j}\nu |>\lambda \right]\ge \frac{C\parallel \nu \parallel }{\lambda },\phantom{\rule{3.33333pt}{0ex}}\lambda >0$

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

@article{AIF_1981__31_1_257_0,
author = {Varopoulos, Nicolas Th.},
title = {A theorem on weak type estimates for Riesz transforms and martingale transforms},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {31},
number = {1},
year = {1981},
pages = {257-264},
doi = {10.5802/aif.826},
zbl = {0437.60003},
mrnumber = {84e:60070},
language = {en},
url = {http://www.numdam.org/item/AIF_1981__31_1_257_0}
}

Varopoulos, Nicolas Th. A theorem on weak type estimates for Riesz transforms and martingale transforms. Annales de l'Institut Fourier, Volume 31 (1981) no. 1, pp. 257-264. doi : 10.5802/aif.826. http://www.numdam.org/item/AIF_1981__31_1_257_0/

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