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

Abstract

Suppose that f is a function on R n such that exp(a|·| 2 )f and exp(b|·| 2 ) ˆ f are bounded, where a,b > 0. Hardy's Uncertainty Principle as- serts that if ab > � 2 , then f = 0, while if ab = � 2 , then f = cexp( a|·| 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.