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Mat. Zametki, 2013, Volume 94, Issue 1, Pages 3–21 (Mi mz10271)  

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

F. Abdelmoula, A. Baklouti, D. Lahyani

University of Sfax

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 $2\leq p,q\leq +\infty$. 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, Fourier transform, Plancherel formula.

DOI: https://doi.org/10.4213/mzm10271

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English version:
Mathematical Notes, 2013, 94:1, 3–19

Bibliographic databases:

UDC: 517
Received: 20.03.2013

Citation: F. Abdelmoula, A. Baklouti, D. Lahyani, “An $L^p$$L^q$ Analog of Miyachi's theorem for Nilpotent Lie Groups and Sharpness Problems”, Mat. Zametki, 94:1 (2013), 3–21; Math. Notes, 94:1 (2013), 3–19

Citation in format AMSBIB
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