The $L^p$–$L^q$ analog of Morgan's theorem on exponential solvable Lie groups

In this paper, we define an analog of the $L^p$–$L^q$ Morgan's uncertainty principle for any exponential solvable Lie group $G(p,q\in[1,+\infty])$. When G is nilpotent and has a noncompact center, the proof of such an analog is given for $p,q\in[2,+\infty]$, extending the earlier settings ([2], [4] and [5]). Such a result is only known for some particular restrictive cases so far. We also prove the result for general exponential Lie groups with nontrivial center.