Sharp subelliptic estimates in the $\bar{\mathrm{\partial }}$-Neumann problem via an uncertainty principle

The problem of giving a (CR-)geometric description of the best possible order of a subelliptic estimate at a boundary point in the $\bar{\mathrm{\partial }}$-Neumann problem is largely open. In this paper, we introduce a novel technique based on a “$\bar{\mathrm{\partial }}$-uncertainty principle” and, as an application, we determine the sharp order of subellipticity at the origin for a large class of Kohn’s special domains in ambient dimension $\le 5$.