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

Varopoulos, Nicolas Th.

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

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

$$m\left[\sum _{j=1}^n|R_j\nu |>\lambda \right]\ge \frac{C\parallel \nu \parallel }{\lambda },\lambda >0$$

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

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

$$m\left[\sum _{j=1}^n|R_j\nu |>\lambda \right]\ge \frac{C\parallel \nu \parallel }{\lambda },\lambda >0$$

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

[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