Vector-valued distributions and Hardy's uncertainty principle for operators
Let \(f\) be a function on \(\mathbb{R}^n\) such that \(\text{exp}(a |\cdot |^2) f\) and \(\text{exp}(b |\cdot |^2) \hat f\) are bounded, where \(a,b>0\) and \(\hat f\) denotes the Fourier transform of \(f\). According to \textit{G. H. Hardy}'s Uncertainty Principle [J. Lond. Math. Soc. 8, 227--231 (1933; Zbl 0007.30403; JFM 59.0425.01)], if \(ab>\pi^2\), then \(f=0\), while if \(ab=\pi^2\), then \(f=c\,\text{exp}(-a |\cdot |^2)\). The paper contains a survey of the most important publications related to this famous result. New interesting generalizations to vector-valued functions and operators are obtained.
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