Let T be a singular integral operator, and let 0 < α < 1. If t > 0 and the functions f and Tf are both integrable, then there exists a function g ε BLipα(ct)) such that ∥f - g∥≤ Cdist L1 (f, BLipα (t)and ∥Tf-Tg ∥ L 1 ≤ c ∥ f-g ∥ L1 +dist L1 (Tf B Lip α(t)). (Here B X (τ) is the ball of radius τ and centered at zero in the space X; the constants C and c do not depend on t and f.) The function g is independent of T and is constructed starting with f by a nearly algorithmic procedure resembling the classical Calderón-Zygmund decomposition.