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 ff is a function on Rn\mathbb{R}^n such that exp(a2)f\exp(a |\cdot|^2) f and exp(b2)f^\exp(b |\cdot|^2) \hat f are bounded, where a,b>0a,b > 0. Hardy's Uncertainty Principle asserts that if ab>π2ab > \pi^2, then f=0f = 0, while if ab=π2ab = \pi^2, then f=cexp(a2)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