Stability of Approximation Under the Action of Singular Integral Operators

Let $T$ be a singular integral operator, and let $0<\alpha<1$. If $t>0$ and the functions $f$ and $Tf$ are both integrable, then there exists a function $g\in B_{\operatorname{Lip}_{\alpha}}(ct)$ such that
$$ \|f-g\|_{L^1}\le C\operatorname{dist}_{L^1}(f,B_{\operatorname{Lip}_{\alpha}}(t)) $$
and
$$ \|Tf-Tg\|_{L^1}\le C\|f-g\|_{L^1}+\operatorname{dist}_{L^1} (Tf,B_{\operatorname{Lip}_{\alpha}}(t)). $$
(Here $B_X(\tau)$ is the ball of radius $\tau$ 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.