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

### Michael Cowling

University of New South Wales, Sydney, Australia### Bruno Demange

Université de Grenoble I, Saint-Martin-d'Hères, France### M. Sundari

Chennai Mathematical Institute, Siruseri, India

## Abstract

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.

## Cite this article

Michael Cowling, Bruno Demange, M. Sundari, Vector-valued distributions and Hardy’s uncertainty principle for operators. Rev. Mat. Iberoam. 26 (2010), no. 1, pp. 133–146

DOI 10.4171/RMI/597