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 νM(R n ) satisfy the weak type inequality

mj=1n|Rjν|>λCνλ,λ>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 νM(R n ) satisfont à l’inégalité faible

mj=1n|Rjν|>λCνλ,λ>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},
     address = {Gap},
     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/

[1] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University press (1970). | MR 44 #7280 | Zbl 0207.13501

[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 0466.31006

[4] L. Loomis, A note on the Hilbert transform, B.A.M.S., 52 (1946), 1082-1086. | MR 8,377d | Zbl 0063.03630

[5] K. Murali Rao, Quasi-Martingales, Math. Scand., 24 (1969), 79-92. | MR 43 #1265 | Zbl 0193.45502

[6] S. Janson, Characterizations of H1 by singular integral transforms on martingales and Rn, Math. Scand., 41 (1977), 140-152. | MR 57 #3729 | Zbl 0369.42005