Uncertainty principle related to Flensted–Jensen partial differential operators

This paper deals with some formulations of the uncertainty principle associated to generalized Fourier transform [Formula: see text] related to Flensted–Jensen partial differential operators. The aim result is to prove the analogue of Bonami–Demange–Jaming’s theorem : A version of Beurling–Hörmander’s theorem which gives more precision in the form of nonzero functions verifying modified-Beurling’s condition. As application, we get analogous of Gelfand–Schilov’s theorem, Cowling–Price’s theorem and Hardy’s theorem for [Formula: see text].

HARDY’S THEOREM FOR GABOR TRANSFORM

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form$\mathbb{R}^{n}\times K$, where$K$is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group$G$which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form$G\times D$, where$D$is a discrete group.

A note on Hardy's theorem

International audience Hardy's theorem for the Riemann zeta-function ζ(s) says that it admits infinitely many complex zeros on the line (s) = 1 2. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.

HARDY AND MIYACHI THEOREMS FOR HEISENBERG MOTION GROUPS

Let $G=\mathbb{H}^{n}\rtimes K$ be the Heisenberg motion group, where $K=U(n)$ acts on the Heisenberg group $\mathbb{H}^{n}=\mathbb{C}^{n}\times \mathbb{R}$ by automorphisms. We formulate and prove two analogues of Hardy’s theorem on $G$. An analogue of Miyachi’s theorem for $G$ is also formulated and proved. This allows us to generalize and prove an analogue of the Cowling–Price uncertainty principle and prove the sharpness of the constant $1/4$ in all the settings.

Some uncertainty principles for diamond Lie groups

So far, the uncertainty principles for solvable non-exponential Lie groups have been treated only in few cases. The first author and Kaniuth produced an analogue of Hardy's theorem for a diamond Lie group, which is a semi-direct product of ℝ