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L^p and weak L^1 estimates for the maximal Riesz transform and the maximal Beurling transform
We prove L p estimates for the maximal Riesz transform in terms of the Riesz transform itself, for 1 < p ≤ ∞.We show that the corresponding weak L 1 estimate fails for the maximal Riesz transform, but surprisingly does hold for the maximal Beurling transform.