Geodesic
An $L^p$--$L^q$ Analog of Miyachi's theorem for Nilpotent Lie Groups and Sharpness Problems
Matematičeskie zametki, Tome 94 (2013) no. 1, pp. 3-21

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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.
Keywords: uncertainty principle
Mots-clés : Fourier transform, Plancherel formula. 
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     author = {F. Abdelmoula and A. Baklouti and D. Lahyani},
     title = {An $L^p$--$L^q$ {Analog} of {Miyachi's} theorem for {Nilpotent} {Lie} {Groups} and {Sharpness} {Problems}},
     journal = {Matemati\v{c}eskie zametki},
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     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2013_94_1_a0/}
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F. Abdelmoula; A. Baklouti; D. Lahyani. An $L^p$--$L^q$ Analog of Miyachi's theorem for Nilpotent Lie Groups and Sharpness Problems. Matematičeskie zametki, Tome 94 (2013) no. 1, pp. 3-21. http://geodesic.mathdoc.fr/item/MZM_2013_94_1_a0/