An $L^p$–$L^q$ Analog of Miyachi's theorem for Nilpotent Lie Groups and Sharpness Problems

The purpose of this paper is to formulate and prove an $L^p$–$L^q$ analog of Miyachi's theorem for connected nilpotent Lie groups with noncompact center for $2\leq p,q\leq +\infty$. This allows us to solve the sharpness problem in both Hardy's and Cowling–Price's uncertainty principles. When $G$ is of compact center, we show that the aforementioned uncertainty principles fail to hold. Our results extend those of [1], where $G$ is further assumed to be simply connected, $p=2$, and $q=+\infty$. When $G$ is more generally exponential solvable, such a principle also holds provided that the center of $G$ is not trivial. Representation theory and a localized Plancherel formula play an important role in the proofs.