$Tb$ criteria for Calderón—Zygmund operators on Lipschitz spaces with para-accretive functions

By developing the Littlewood—Paley characterization of Lipschitz spaces $\operatorname{Lip}(\alpha){(\mathbb{R}^n)}$ and the new Lipschitz spaces $\operatorname{Lip}_b(\alpha){(\mathbb{R}^n)}$ with $b$ a para-accretive function, and establishing a density argument for $\operatorname{Lip}_b(\alpha){(\mathbb{R}^n)}$ in the weak sense, the authors prove that the Calderón—Zygmund operators $T$ are bounded from $\operatorname{Lip}_b(\alpha){(\mathbb{R}^n)}$ to $\operatorname{Lip}(\alpha){(\mathbb{R}^n)}$ if and only if $T(b)=0$.