Stability of approximation under the action of singular integral operators

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 $g \in B_{Lip_\alpha } (ct)$ such that $\left\| {f - g} \right\|_{L^1 } \leqslant Cdist_{L^1 } (f,B_{Lip_\alpha } (t))$ and $\left\| {Tf - Tg} \right\|_{L^1 } \leqslant C\left\| {f - g} \right\|_{L^1 } + dist_{L^1 } (Tf,B_{Lip_\alpha } (t)).$ . (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.