Stability of approximation under the action of singular integral operators (Q883702)

Let \(X=L^p(\mathbb R^n,H),\) \(1<p<\infty,\) or \(X= \text{Lip}_\alpha (\mathbb R^n,H),\) \(0<\alpha <1,\) where \(H\) is an Euclidean space, and let \(T\) be a singular integral operator, bounded on \(X\) and having certain additional properties. Let the function \(f\) be such that \(f, Tf \in L^1(\mathbb R^n,H).\) Using a generalized Calderón-Zygmund decomposition, the authors construct for every \(t>0\) a function \(f_t\) from the ball \(B_X(ct)\) of radius \(ct\), centered at zero in the space \(X\), such that \[ \begin{aligned} \| f-f_t\| _{L^1} &\leq C \text{ dist}_{L^1} (f, B_X(t)), \\ \| Tf-Tf_t\| _{L^1} &\leq C(\| f-f_t\| _{L^1}+ \text{dist}_{L^1} (Tf, B_X(t))). \end{aligned} \]