Revista Matemática Iberoamericana

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Volume 26, Issue 1, 2010, pp. 133–146
DOI: 10.4171/RMI/597

Published online: 2010-04-30

Vector-valued distributions and Hardy’s uncertainty principle for operators

Michael Cowling[1], Bruno Demange[2] and M. Sundari[3]

(1) University of New South Wales, Sydney, Australia
(2) Université de Grenoble I, Saint-Martin-d'Hères, France
(3) Chennai Mathematical Institute, Siruseri, India

Suppose that $f$ is a function on $\mathbb{R}^n$ such that $\exp(a |\cdot|^2) f$ and $\exp(b |\cdot|^2) \hat f$ are bounded, where $a,b > 0$. Hardy's Uncertainty Principle asserts that if $ab > \pi^2$, then $f = 0$, while if $ab = \pi^2$, then $f = c\exp(-a|\cdot|^2)$. In this paper, we generalise this uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.

Keywords: Uncertainty principle, linear operators, Hardy’s theorem

Cowling Michael, Demange Bruno, Sundari M.: Vector-valued distributions and Hardy’s uncertainty principle for operators. Rev. Mat. Iberoamericana 26 (2010), 133-146. doi: 10.4171/RMI/597