Let D be a smoothly bounded domain in \({\mathbb{C}}^ n\), and let S and B denote the Szegö and the Bergman projections for D. The authors prove some estimates under certain technical assumptions on D, which make precise the principle that estimates for S are related to estimates for B. In particular, their results imply that if D is the unit ball, and \(N=\sum z_ j \partial /\partial z_ j\), then \[ \| Su-Bu\|_ s