## Revista Matemática Iberoamericana

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

#### 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.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 1 (2010), 133-146.

Dates
First available in Project Euclid: 16 February 2010

https://projecteuclid.org/euclid.rmi/1266330120

Mathematical Reviews number (MathSciNet)
MR2666311

Zentralblatt MATH identifier
1195.42050

#### Citation

Cowling, M. G.; Demange, B.; Sundari, M. Vector-valued distributions and Hardy's uncertainty principle for operators. Rev. Mat. Iberoamericana 26 (2010), no. 1, 133--146. https://projecteuclid.org/euclid.rmi/1266330120

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