In this talk, we obtain a complete characterization of the Carleman inequality with the weights $e^{v\cdot x}$ for the polyharmonic operators and deduce unique continuation properties for higher order Schrödinger operators with critical potentials. Closely tied toolkits from harmonic analysis are the Stein–Tomas restriction estimate and the endpoint estimates for the Bochner–Riesz operators of negative indices. This is based on joint work with Eunhee Jeong and Sanghyuk Lee.