An L p -L q analog of miyachi’s theorem for nilpotent lie groups and sharpness problems

Abstract

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 ≤ p, q ≤ +∞. 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 = +∞. 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.